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ELEMENTS 

OF   THE 

DIFFERENTIAL  AND  INTEGRAL 
CALCULUS 


Wiitb  Bjamples  auD  practical  Hpplications 


BY  ^ 

J.  W.   NICHOLSON,   A.M.,  LL.D. 

President  and  Professor  of  Mathematics 
Louisiana  Slate  University  and  Agricultural  and  Mechanical  College 


^^TOi^  cor  r  ^ 


MATH.  CTEPT. 


NEW  YORK  AND  NEW  ORLEANS 
UNIVERSITY   PUBLISHING  OOMPANr 

1896 


Copyright,  1896, 

BY 

UNIVERSITY  PUBLISHING  CO. 


All  Bights  Reserved 


***  1673 


150255 


PREFACE. 


If  many  respects  this  work  is  quite  different  from  any  other 
on  the  same  subject,  though  in  preparing  it  there  has  been  no 
attempt  at  originality  beyond  presenting  the  principles  in  a  more 
tangible  form  than  usual,  and  thns  securing  a  better  text-book 
for  the  ordinary  student  of  mathematics.  The  aim  has  been 
to  prepare  a  work  for  beginners,  and  at  the  same  time  to  make 
it  sufficiently  comprehensive  for  the  requirements  of  the  usual 
undergraduate  course. 

The  chief  distinction  of  the  treatise  is  that  it  is  based  on  the 
conception  of  Projyortional  Valuations.  This  method  has  been 
employed  as  the  most  elementary  and  practical,  and  none  the 
less  rigorous  or  general,  form  of  presenting  the  first  principles  of 
the  subject  (see  the  following  Note). 

Diiferentiation  and  Integration  are  carried  on  together,  and 
the  early  introduction  of  practical  applications  both  of  the  dif- 
ferential and  integral  calculus,  which  this  mode  of  presenting 
the  subject  permits,  is  intended  to  serve  an  important  purpose  in 
illustrating  the  utility  and  potentiality  of  the  science,  and  arous- 
ing the  interest  of  the  student.  The  formulas  for  differentiating 
and  integrating  are,  as  a  rule,  expressed  in  terms  of  tt  and  v 
instead  of  x,  u  and  v  being  functions  of  x.  The  advantages  thus 
secured  are  obvious. 

Among  the  additional  features  of  special  interest  may  be 
mentioned  the  following  :  (1)  The  treatment  of  ffeas  a  variable 
indej)endent  of  x  (Art.  68,  and  Appendix,  A3);  (2)  a  rigorous 
deduction  of  a  simple  test  of  absolute  convergency,  without 
recourse  to  the  remainder  in  Taylor's  formula  (Arts.  115  to  119) ; 
(3)  an  extension  of  the  ordinary  rules  for  finding  maxima  and 
minima  (Arts.  140  to  143)  ;  (4)  a  chapter  on  Independent 
Integration   (Chap.  IX);  (5)  integration   by  indeterminate  co- 


iv  PREFACE. 

efficients  (Arts.  211  to  216);  (6)  the  introduction  of  turns  in 
curve-tracing  (Arts.  175  to  179) ;  and  (7)  a  new  proof  of  Taylor's 
formula,  which  is  believed  to  be  as  rigorous  as,  and  less  artificial 
than,  those  in  general  lise  (Appendix,  A  J. 

In  prej)aring  the  book  the  best  available  authors  have  been 
consulted,  and  many  of  the  examples  have  been  taken  from  the 
works  of  Todhunter,  Williamson,  Courtenay,  Byerly,  Eice  and 
Johnson,  Taylor,  Osborne,  Loomis,  and  Bowser. 

I  improve  this  opportunity  to  tender  my  thanks  to  Prof. 
William  Hoover  of  the  Ohio  University,  Prof.  Alfred  Hume 
of  the  University  of  Mississippi,  and  Prof.  0.  D.  Smith  of  the 
Polytechnic  Institute  of  Alabama  for  valuable  assistance  in  the 
reading  of  proofs.  Their  corrections  and  suggestions  have 
relieved  the  treatise  from  various  imperfections  it  would  other- 
wise have  contained. 

Further  acknowledgments  of  indebtedness  are  also  due  to 
my  colleague.  Prof.  C.  Alphonso  Smith,  of  the  Department  of 
English,  who  has  aided  me  with  his  scholarly  criticisms. 

James  W.  N'icHOLSoiir. 
Baton  Rouge,  La.,  1896. 


Note. — Tlie  metliod  of  Proportional  Variations,  wliicli  is  the  suggestion 
and  outgrowth  of  work  in  the  class-room,  is  believed  to  possess  the  follow- 
ing merits  : 

{a)  The  conception  is  one  with  which  the  student  is  already  familiar, 
for  the  principle  of  proportional  changes  is  among  the  first  that  he  encoun- 
ters, even  in  the  lower  mathematics. 

(p)  It  affords  finite  differentials,  and,  without  introducing  infinitesimals, 
or  infinitely  small  quantities,  or  "the  foreign  element  of  time,"  has  all  the 
advantages  of  the  differential  notation. 

(c)  In  many  cases  the  proportional  variations  (or  differential)  can  be 
detected  by  inspection  (see  Arts.  31,  32,  35),  and  in  all  cases  they  may  be 
deduced  b_y  the  theory  of  limits.  Hence  the  method  has  all  the  lucidity  of 
finite  differences  and  all  the  rigor  of  the  doctrine  of  limits. 

{d)  It  is  a  method  to  which  the  doctrines  of  Infinitesimals  and  Rate  of 
Change  are  easy  corollaries. 

{e)  In  general,  the  form  and  properties  of  the  increments  of  all  quan- 
tities are  due  to  proportion  and  acceleration,  or  to  proportional  and  dispro- 
portional  changes  ;  hence,  a  system  of  Calculus  based  on  such  changes 
adapts  itself  naturally  to  questions  in  Geometry,  Mechanics,  and  Physics. 


CONTENTS. 


CHAPTER   I. 

FUNDAMENTAL  PRINCIPLES. 
[Pages  1  to  20.] 

Quantity.  Variables  and  constants.  Art.  1.*  Dependent  and  indepen- 
dent variables,  2.  Functions,  3.  Increasing  and  decreasing  functions,  4. 
Explicit  and  implicit  functions,  5.  Algebraic  and  Transcendental  func- 
tions, 6.     Continuous  functions,  7.     Notation  of  functions,  8.     Examples. 

Increments.  Increments  of  independent  variable  and  function,  nota- 
tion,  and  illustrations,  9.     General  formula,  10.     Examples. 

Variation.  Proportional  variation,  11.  Principles,  13  to  15.  Dispro- 
portional  variation,  16.  Principles,  17  to  19.  Composition  of  increments, 
illustrations,  20. 

Theory  of  Limits.  Definition  and  illustrations  of  limits.  21.  Prin- 
ciples, 22,  23.  Proof  of  the  formula,  Jy  =  mji  -\-  WaA'^,  and  geometrical 
illustration,  24. 

Diflferentials  and  Accelerations.  Definition  and  notation  of  differ- 
ential and  acceleration,  25.  Corollaries.  Derivatives,  26.  How  the  pro- 
portional variations  or  differential  of  y  =f{.r)  may  be  found,  27  to  29. 

Differentials  of  Geometric  Functions.  Differentiation,  30.  Plane 
areas  in  rectangular  co-ordinates,  31.  Solids  of  revolution,  32.  Arcs  of 
curves  in  rectangular  co-ordinates,  33.  Surfaces  of  revolution,  34.  Plane 
areas  in  polar  co-ordinates,  35. 

CHAPTER   II. 

ELEMENTARY  DIFFERENTIATION  AND  INTEGRATION. 
[Pages  21  to  42.] 

Differentiation.  Rules  for  differentiating,  36.  Differential  of  o  or  G, 
37.     Differential  of  CT,  38.     Differential  of  vy,  39.     Differential  of  vyz,  40. 

*  Similarly,  the  following  numbers  refer  to  articles  and  not  to  rages. 

V 


vi  CONTENTS. 


DifEerential  of  -,  41  ;  also  of  -,  42.     Differential  of  «",  43  ;  also  of  i/«,  44. 

y  y 

Differential  oiv-^-y  —  z,  45.     Examples, 

Slope  of  Curves.  Direction  and  slope  of  a  line,  46.  Direction  of  a 
curve  at  any  point,  47.  Slope  of  a  curve,  48.  tan  (p,  sin  0,  cos  (p,  wiiere 
<p  is  the  angle  of  direction  of  a  curve,  49.     Examples. 

lutegration.  Definition  and  sign,  50.  Dependent  integration,  51. 
Integral    of   0,    52.       Integral    of    cd'o,    53.      Integral    of    V'dv,    54,    55. 

/  v'^d'o  =  -  I  v^cdv,  56.     Integral  of  (a  -|-  hx'")Px''-'^dx,  57.     Integral  of 

du  -\-  d'd  —  dz,  58.     Examples. 

Problems.  The  problem  of  integration,  59.  Definite  value  of  the  con- 
stant G,  60.  Examples.  Applications  to  geometry,  61.  Areas  of  curves, 
62.  Definite  integrals,  63.  Length  of  curves,  64.  Areas  of  surfaces  of 
revolution,  65.     Volumes  of  solids  of  revolution,  66. 

CHAPTER  III. 

SUCCESSIVE  DIFFERENTIALS  AND  RATE  OF  CHANGE. 
[Pages  43  to  54.] 

Successive  Differentials.  Definition  and  Notation,  67.  Why  dx  may 
be  treated  as  a  constant  in  successive  differentiation,  68.  Examples. 
Liebnitz's  theorem,  69. 

Kate  of  Change.  Uniform  change,  70.  Variable  change,  71.  Ex- 
amples. Applications  to  geometry,  72.  Given  the  rate  of  change  of  a  curve 
to  find  the  rates  of  change  of  its  co  ordinates,  73. 

Application  to  3Ieclianics.  Velocity,  74.  Examples.  Positive  and 
negative  velocity,  75.  Examples.  Uniformly  accelerated  motion,  76. 
Formulas  for  the  free  fall  of  bodies  in  vacuo,  77.     Problems  in  Mechanics. 

CHAPTER   IV. 

GENERAL  DIFFERENTIATION. 

[Pages  55  to  83.] 

/  1\^ 

LiOgarithms.     Lemma,  The  limit  of     1  -| —  1  ,  as  s  approaches  oo ,  78. 

Differential  of  loga  v,  79.    Examples. 

Exponential  Functions.  The  differential  of  a^,  80.  Examples. 
Differential  of  ?/^,  81.     Examples. 

Trigonometric  .  Functions.  Circular  measure  of  angles,  82.  Dif- 
ferential of  sin  V  and  cos  v,  83.  Sin  dv  =  dv,  cos  dv  =  1,  84.  Differential 
of  tan  V,  85.  d{cot  «),  86.  d{seG  v),  87.  d(co'sec «),  88.  c?(vers  »),  89. 
Examples. 


CONTENTS.  Vll 

Inverse  Trigonometric  Functions,  (^(sin-i  y),  90.  r?(cos-i  y),  91. 
(Z(tan-i  j^),  92.  (Z(cot-i  ?/),  93.  (?(sec-i  y),  94.  (Z(cosec-i  j'),  95.  (^(vers-i?/), 
96.  Examples.  Differential  of  an  arc  in  polar  co-ordinates,  97.  tan  ^. 
sin  ip,  cos  ip,  where  ip  is  the  angle  formed  by  the  tangent  and  radius  vector, 
98. 

Functions  of  two  or  more  Independent  variables.  How  such 
functions  may  vary,  99.  A  partial  differential,  100.  Total  differential, 
101,  102.  A  partial  derivative,  103.  The  total  derivative,  104.  Examples. 
Function   of  Functions,   105.     Examples.     Successive  partial  differentials 

and  derivatives,   106.     - — r-  =  - — — ,  107.     Examples.     To  find  the   suc- 
ax  ay      dy  dx 

cessive  differentials  of  a  function  of  two  independent  variables,  108.  Im- 
plicit functions,  109.  Examples.  Successive  derivatives  of  an  implicit 
function,  110.     Examples.     Change  of  the  independent  variable,  111.     To 

find  the  successive  derivatives  of  —  when  neither  x  nor  y  is  independent, 

112.     Examples.     Miscellaneous  Examples. 

CHAPTER   V. 

SERIES,  DEVELOPMENT  OF  FUNCTIONS,    AND   INDETERMINATE   FORMS. 

[Pages  84  to  105.] 

Series.  Definition,  113.  To  develop  a  function,  114.  An  absoh^tely 
convergent  series,  115.  Test  of  absolutely  convergent  series,  116.  Coral- 
laries,  117  to  119.     Examples. 

Development  of  Functions.  Two  formulas,  120.  Taylor's  formula, 
■'21,  122.  Binomial  theorem,  123.  Developments  of  sin  (y -\- x),  sin  x, 
cos  X,  cos  (y  -\-  x),  124.  Development  of  log  (y  +  x),  125.  Maclaurin's 
formula,  126.  Developments  of  a^,  e^,  e,  127.  Development  of  tan-i  x, 
128.  Examples.  To  find  the  value  of  tt,  129.  To  compute  natural  loga- 
rithms, 130.     To  compute  common  logarithms,  131 . 

0 
Indeterminate  Forms.     How  the  form  —  arises,  132.     To   evaluate 

functions  of  the  form —,  133.    Examples.    Of  the  form —,  134.     Examples. 
0  00 

Of  the  forms  0  X  <»  and  oo  —  oo ,  135.     Examples.     Of  the  forms  0°,  oo  °, 

and  1   ,  136,     Examples.     Implicit  functions,  137.     Examples. 

CHAPTER  VI. 

MAXIMA  AND  MINIMA. 

[Pages  106  to  121.] 

Definitions  and  Principles,  Definitions  and  illustrations,  138.  If 
f(^a')  is  a  max.  or  min.  then  f'{a')  =  0  or  oo,  139.    /(a)  is  neither  a  max. 


Yin  COI^TENTS. 

noramin.,  if  an  even  number  of  tlae  roots  of /'(a")  =  0  and/(a")  =:  co  are  equal 
to  a' ,  140.  /(a')  is  a  max.  or  min.  if  an  odd  number  of  tbe  roots  of 
f{x)  =  0  and /(a')  =  oo  are  equal  to  a' ,  141.  Max.  and  min.  occur  alternately, 
142. 

Rules  for  Finding  Maxima  and  Minima.  When  all  tlie  roots  of 
f{x)  =  0  and  co  are  known  or  can  be  conveniently  found,  143.  I.  By  sub- 
stituting a'  —  h  and  a'  -{-Ji  fox  x,  144.  II.  By  Taylor's  formula,  145.  Self- 
evident  principles  which  serve  to  facilitate  the  solution  of  problems,  146. 
Examples  and  problems. 

Functions  of  two  Independent  Varial>les.  Definition,  147.  Con- 
ditions for  maxima  and  minima,  148.     Examples. 

CHAPTER   VII. 

APPLICATIONS  OF  THE  DIFFERENTIAL  CALCULUS    TO  PLANE  CURVES. 

[Pages  183  to  159.] 

Tangents,  Normals,  and  Asymptotes.  Equations  of  the  tangent 
and  normal,  149.  Lengths  of  tangent,  normal,  subtangent,  and  subnormal, 
150.  Examples.  Lengths  of  tangent,  normal,  subtangent,  and  subnormal 
in  polar  co-ordinates,  151.  Examples.  Asymptote,  152.  General  equa- 
tion of  an  asymptote,  153.  Relation  of  2/  to  a;  when  they  are  infinite,  154. 
Examples.     Asymptotes  determined  by  inspection,  155.     Examples. 

Curvature.  Total  curvature,  156.  Uniform  curvature,  157.  Variable 
curvature,  158.  Radius  of  curvature,  159.  Examples.  Radius  of  curva- 
ture in  polar  co-ordinates,  160.     Examples. 

Contact  of  DiflPerent  Orders.  Definitions,  161.  When  two  curves 
cross  or  do  not  cross  at  their  point  of  contact,  162.  Examples.  Osculating 
curves,  163.     Osculating  straight  line,  164.     Osculating  circle,  165. 

Involutes  and  Evolutes.  Definitions,  166.  Elementary  Principles, 
167.     To  find  the  equation  of  the  evolute  of  a  given  curve,  168.     Examples. 

Envelopes.  Definition,  169.  The  envelope  is  tangent  to  every  curve 
of  the  series,  170.  To  find  the  equation  of  the  envelope  of  a  given  series 
of  curves,  171.     Examples. 

Tracing  Curves.  The  general  form  of  a  curve,  etc.,  172.  Direction 
of  curvature,  convex  and  concave  arcs,  173.  Point  of  inflection  and  prin- 
ciples, 174.     Examples. 

Singular  Points.  Definition  (etc.).  cc-turns  and  ^/-turus,  and  multiple 
points,  175.  To  determine  the  positions  of  the  singular  points  of  curves, 
176.  To  determine  tlie  character  of  the  multiple  points  of  curves,  177. 
Examples.  Tracing  polar  curves,  178.  Examples.  The  character  of 
multiple  points  often  more  easily  determined  by'changing  to  polar  co  ordi- 
nates,  179.     Examples. 


CONTENTS.  IX 

CHAPTER   VIII. 

GENERAL  DEPENDENT  INTEGRATION. 
[Pages  160  to  190.] 

Fundamental  Formulas.     Twenty-two  formulas,  180. 

Reduction  and  Integration  of  Differentials.  Reduction  of  Differ- 
entials, definition,  and  how  effected,  181.  By  constant  multipliers,  182. 
Examples. 

Reduction  of  Differentials  by  Decomposition.  How  effected,  183. 
Elementary  differentials,  184.  Examples.  Trigonometric  differentials,  185. 
Examples.  How  trigonometric  differentials  may  often  be  more  conveniently 
integrated,  186.  Rational  fractions,  187.  AVhen  the  simple  factors  of  the 
denominator  are  real  and  unequal,  188.  Examples.  When  some  of  the 
simple  factors  of  the  denominator  are  real  and  equal,  189.  Examples. 
When  some  of  the  factors  of  the  denominator  are  imaginary  and  unequal, 
190.  Examples.  When  some  of  the  simple  factors  of  the  denominator 
are  imaginary  and  equal,  191.     Example. 

Reduction  and  Integration  by  Substitution.  Irrational  differen- 
tials, 192.     Examples.     When  a  -{-  bx  is  the  only  part  having  a  fractional 


exponent,  193.  Examples.  When  4/a  -\-  bx  -\-  x-  or  ^a  -\-  bx  —  x"-  is  the 
only  surd  involved,  194.  Examples.  Binomial  differentials,  195.  Con- 
ditions of  integrability,  196.     Examples. 

Integration  by  Parts.     Fundamental  formula,  197.     Examples. 

Reduction  Formulas.  Definition,  198.  Reduction  formula  fc/ 
xPlog"xdx,ld9.  Examples.  Reduction  formula  for  rt^a;"(?a!,  200.  Examples. 
Reduction  formulas  for  x'^cosaxdx  and  x"  sin  ax  clx,  201.  Examples. 
Reduction  formula  for  X  sin-i  xdx,  202.     Examples. 

Integral  of       ,    "  ,  203.     Integral  of  -^^  and  — ,  204. 

°  a  -[-  &  cos  x  sin  x  cos  x 

Approximate  Integration.     Last  resort  in  separating  a  differential 

into  its  integrable  parts,  205.     Examples.     Development  of   functions  by 

exact  and  approximate  integration,  206.     Examples. 

CHAPTER   IX. 

INTEGRATION  CONTINUED. 
[Pages  191  to  210.] 
Independent  Integration.     Increments   deduced   from   differentials, 
207.     Examples.     Increments   as  definite   integrals,    208.      Examples.     A 
more  convenient  series,  209.     Examples.     Bernouilli's  series,  210. 

Integration  by  Indeterminate  Coefficients.  Explanation  and  formu- 
la, 211.  When  ^  =  0  or  the  integration  is  independent,  212.  Illustra- 
tions and  examples.     Application  to  /sin™  a;  cos"  a;  dr,  213.     When  k  is  not 


X  CONTENTS. 

=  0  or  when  the  integration  is  partly  dependent,  214.  Illustrations  and 
examples.  Reduction  formulas  for  binomial  differentials,  215.  Approx- 
imate integration  and  the  elliptic  differential,  216. 

CHAPTER   X. 

INTEGRATION  AS  A  SUMMATION  OF  ELEMENTS. 
[Pages  211  to  246.] 

Elements  of  Functions.  Differentials  may  be  as  small  as  we  please, 
217.  Elements,  218.  Signification  of  a  definite  integral  as  a  sum,  219. 
Illustrative  examples.  When  dx  is  not  a  constant,  220.  Signification  of  a 
definite  integral  as  the  limit  of  a  sum,  221.  Illustrative  example.  Inte- 
gration equivalent  to  two  distinct  operations,  222. 

Application  to  Geometry,  Length  of  curves,  rectangular  co-ordi- 
nates, 223.  Examples.  Polar  co-ordinates,  224.  Examples.  To  find  the 
equation  of  a  curve  when  its  length  is  given,  225.  Areas  of  curves,  rect- 
angular co-ordinates,  226.  Examples.  Generatrix  of  area,  227.  Polar 
co-ordinates,  228.  Examples.  Areas  of  surfaces  of  revolution,  229.  Ex- 
amples.    Volumes  of  solids  of  revolution,  230. 

Successive  Integration.  A  double  integral,  231.  Definite  double 
integrals,  232.     A  triple  integral,  233.     Examples. 

Areas  of  Surfaces.  Plane  surfaces,  rectangular  co-ordinates,  234. 
Polar  co-ordinates,  235.     Examples.     Surfaces  in  general,  286.     Examples. 

Volumes  of  Solids  Determined  by  Triple  Integration.  Formula, 
237.     Examples. 

Application  to  Mecbanics.     "\'\'OEK,  how  computed,  238.     Example. 

Centre  of  Gravity.  Centre  of  gravity,  moments,  etc.,  239.  Centre  of 
gravity  of  a  plane  area,  240.  Centre  of  gravity  of  a  plane  curve,  241. 
Centres  of  gravity  of  solids  and  surfaces  of  revolution,  242.     Examples 

APPENDIX. 

[Pages  247  to  256.] 

Differentiable  functions,  Ai.  Another  illustration  of  the  formula 
/]y  =  mJi-{- viih^,  Ai.  The  differential  of  an  independent  variable  is,  in 
general,  a  variable,  A3.  Another  method  of  finding  the  differentials  of  a'" 
and  logd  V,  A4.  A  rigorous  proof  of  Taylor's  formula,  A5.  Completion  of 
Maclaurin's  formula.  Ae.  The  values  of  Wi  and  «?2  in  the  formula 
Jy  =  nhih  -\-  yn^h^,  A7. 


LIMITED    COTJESES. 

(a)  The  first  three  chapters.  This  course  is  complete  as 
far  as  it  goes,  since  differentiation  and  integration  are  carried  on 
together.  It  embraces  the  notation,  fundamental  principles,  and 
some  of  the  most  important  applications  of  the  Calculus.  The 
student  who  understands  elementary  algebra  and  geometry,  and 
the  construction  of  elementary  loci,  should  find  but  little  dif- 
ficulty in  mastering  it. 

(b)  The  first  six  chapters.  This  adds  to  the  former  course 
the  transcendental  functions,  development  of  functions,  evalua- 
tion of  the  indeterminate  forms,  and  maxima  and  minima. 

Suggestions.  (1)  It  is  recommended  to  omit  the  more 
difficult  examples  and  problems  in  passing  over  the  book  the 
first  time. 

(2)  Aj  of  the  Appendix  may  be  substituted  for  Arts.  116  to 
122,  at  the  discretion  of  the  teacher. 


DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 


CHAP  TEE   I. 
FUNDAMENTAL   PRINCIPLES. 

QUANTITY. 

1.  There  are  two  kinds  of  quantities  employed  in  Calculus, 
variables  and  constants. 

Variables  are  quantities  whose  values  are  to  be  considered  as 
changing  or  cliangeable.  They  are  usually  represented  by  the 
final  letters  of  the  alphabet. 

Constants  are  quantities  whose  values  are  not  to  be  consid- 
ered as  changing  or  changeable.  They  are  usually  represented 
by  the  first  letters  of  the  alphabet.  Particular  values  of  vari- 
ables are  constants. 

2.  Dependent  and  Independent  Variables.  A  Depend- 
ent Variable  is  one  that  depends  upon  another  variable  for 
its  value,  and  an  Independent  Variable  is  one  that  does  not 
depend  on  another  variable,  but  one  to  which  any  arbitrary 
value  or  change  of  value  may  be  assigned.  In  the  elementary 
differential  and  integral  calculus,  the  independent  variable  is 
usually  restricted  to  real  values. 

Thus,  in  m  =  a^  —  7a;  +  5,  v  =  {I  —  x'Y,  y  =  log  (1  +  x), 
u,  V  and  y  are  dependent  variables,  since  they  depend  on  the 

1 


S  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

variable  x  for  their  values;  but  x  is  au  iudependent  variable, 
since,  as  we  may  suppose,  any  value  may  be  assigned  to  it  without 
reference  to  any  other  variable. 

3.  Functions.  Dependent  variables  are  usually  called  func- 
tions of  the  variables  on  which  they  depend.  Hence,  one  vari- 
able is  a  function  of  another  when  the  first  depends  upon  the 
second  for  its  value,  or  Avhen  the  two  are  so  related  that  changes 
in  the  value  of  the  latter  produce  changes  in  the  value  of  the 
former. 

Thus,  the  m^ea  of  a  varying  square  is  a  function  of  its  side', 
the  cost  of  cloth  is  a  function  of  the  quality  and  quantity ;  the 
space  described  by  a  falling  body  is  a  function  of  the  time',  every 
mathematical  expression  depending  on  x  for  its  value,  as  a;% 
(3a;  —  7)',  5a;^  —  6a;  +  11,  etc.,  is  a  function  of  x. 

4.  Increasing  and  Decreasing  Functions.  An  Increasing 
Function  is  one  that  increases  when  the  variable  increases, 
as  {x-\-iy,  '6x^,  log  (5  -\-x)',  and  a  Decreasing  Function  is  one 


that  decreases  when  the  variable  increases,  as  V'lO  —  a;%  -,  etc. 

A  function  of  x  may  be  increasing  for  certain  values  of  x, 
and  decreasing  for  other  values. 

Thus,  y  ^  x^  —  4x  +  5  is  a  decreasing  function  for  all  values 
of  a;  <  2,  but  increasing  for  all  values  of  x  >  2. 

5.  Explicit  and  Implicit  Functions.  An  Explicit  Function 
is  one  whose  value  is  directly  expressed  in  terms  of  the  variable 
and  constants. 

Thus,  in  the  equations  y  =  [a  —  xY,  y  =  x'  -\-  3a;  +  5,  ^  is 
an  explicit  functix>n  of  x. 

An  Implicit  Function  is  one  whose  value  is  implied  in  an 
equation,  but  not  expressed  directly  in  terms  of  the  variable  and 
constants. 

Thus,  in  the  equation  x^  +  2,xy  -\-  by  —  10,  y  is  an  implicit 
function  of  x,  and  x  is  an  implicit  functioli  of  y.  By  solving  the 
equation  for  x  or  y,  the  function  becomes  explicit. 


FUNDAMENTAL  FBINCIPLE8.  3 

6.  Algebraic  and  Transcendental  Functions.  One  variable 
is  called  an  Algebraic  Function  of  another  when  the  two 
are  connected  by  an  algebraic  equation;  that  is,  an  equation 
which  contains  a  finite  number  of  terms  involving  only  constant 
integral  powers  of  the  variables,  or  an  equation  which  admits  of 
being  reduced  to  this  form. 

Thus,  in  y  =  x^  —  5x,  or  x'l/^  —  xy^  +  ^^V  —  5  =  0,  or 
yl  —  Vax^  +  ?/  =  7,  y  is  an  algebraic  function  of  x,  and  vice 
versa. 

If  two  variables  are  connected  by  an  equation  which  is  not 
algebraic,  each  is  called  a  Transcendental  Function  of  the  other. 

Thus,  if  ^  =  sin  x,  ?/  is  a  transcendental  function  of  x,  and 
x  of  y. 

The  following  are  the  elementary  transcendental  functions: 

A  Logarithmic  Function  is  one  that  involves  the  logarithm 
of  a  variable;  as,  log  x,  log  {a  +  y)- 

An  Exponential  Function  is  one  in  which  the  variable 
enters  as  an  exponent;  as,  «%  y^. 

A  Trigonometric  Function  is  the  sine,  cosine,  tangent,  etc., 
of  a  variable  angle;  as,  sin  x,  cos  y. 

An  Inverse-Trigonometric  Function  is  an  angle  whose 
sine,  cosine,  tangent,  etc.,  is  a  variable;  as,  sin~^a;,  cos~\?/, 
tan~^  t,  etc.,  which  are  read,  "an  angle  whose  sine  is  x,"  "an 
angle  whose  cosine  is  y,"  etc^ 

7.  Continuous  Functions.  A  function  of  a  variable  is  con- 
tinuous between  certain  values  of  the  variable  (1)  when  it  has 
a  finite  value  for  every  value  of  the  variable,  and  (2)  when  the 
changes  in  its  value  corresj)onding  to  indefinitely  small  changes 
in  the  value  of  the  variable  are  themselves  indefinitely  small. 

Thus,  m  y  =  ax  -\-  h,  or  y  =  sin  x,  or  y  —  e^,  y  is  continu- 
ous for  all  finite  real  values  oi  x;  so  also  in  ?/  =  Va''  —  x\  but 
as  a  real  quantity  only  for   real  values   of   x  y  —  a  and  <  a. 

Again,  y  = ^  is  not  continuous  between  the  limits  a;  =  1 

and  X  =  3,  for  when  x  =  2,  y  =  co . 


4  DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

8.  Notation  of  Functions.  The  symbol  f{x)  is  used  to 
denote  any  function  of  x,  and  is  read,  "  function  of  x."  To  de- 
note different  functions  of  x  we  employ  other  symbols,  as  F{x), 
f'{x),  <p{x),  0{x),  etc.  According  to  this  notation,  y  =f{x)  rep- 
resents any  equation  between  x  and  y  when  solved  for  y. 

Thus,  solving  the  equation  y'^  —  2axy  -\-  bx^  =  0  for  y,  we 
obtain  y  =  ax  ±   Vtfx^  —  bx^,  or  y  =f{x). 

The  result  of  substituting  any  number,  as  m,  for  x  in  f{x)  is 
denoted  hj  f{m). 


Thus, 

itfix) 

=  x'  - 

5.T  + 

6, 

/(O) 

=  0^- 

-  5 

0  +  6  = 

:6, 

/(I) 

=  1^  - 

-  5 

1  +  6  = 

O 

/(2) 

=  r  - 

-  5 

3  +  6  = 

0, 

/(3) 

=  3^- 

-  5 

3  +  6  = 

0, 

/(4) 

etc. 

-  5 

4  +  6  = 
etc. 

2 

In  f{x)  if  x  be  increased  by  h   the  result  is   denoted  by 

f{x  +  n). 

Thus,  if /(:c)  =  x"  +  5a-,    then 

f{x  +  70  =  (a;  +  70^  +  b{x  +  70 

=  a;'  +  5»  +  (2x  +  5)/i  +  /^\ 


EXAMPLES. 

1.  In  the  function  f{x)  =  x"  —  9x  +  14,  (1)  which  are  the 
constants?  (2)  Which  is  the  variable  ?  (3)  Find  the  values  of 
/(0),/(3),/(7).     (4)  Which  is  the  least:  /(3),/(5)  or/(6)  ? 

Ans.  (1)  9  and  14;  (2)  x';  (3)  14,  0,  o";  (4)/(5). 

2.  Given  f{x)  =  x' —  lOx  +  24.;  (1)  show  that  /(3)  -/(7) 
=  0;  (2)  that /(5)  </(4);  (3)  that /(- 1)  =/(ll);  (4)  that 
f{x  +  70  =x'  -  lO.r  -f-  24  +  {2x  -  10)h  +  h\ 


FUNDAMENTAL  PRINCIPLES.  5 

3.  Eeduce  2a;  +  3?/  +  12  =  0  to  the  form  y  =  f{x). 

ij=-  -K12  +  2x). 

4.  Eeduce  x"  -{-  y""  =  K'  to  the  form  y  =f{x). 

y  =  ±  VK'  —  x\ 

5.  Given /(rr)  =  -  4  -  fa;;  show  that/(0)  -/(3)  =  2. 

6.  Given /(.r)  =  V'imx;  show  that  /{iin)  —  f{m)  =  2m. 

7.  Uf{x)  =  4/100  -  x%  show  that/(6)  =/(8)  +  2. 

8.  In  a''y'  -\-  Ifx^  =  a'h^,  is  ?/  a  function  of  x  ?   Why  ?    What 
function  ? 

(1)  It  is.     (2)  Because  any  change  in  the  vahie  of  x  pro- 
duces a  change  in  the  vakie  of  y.     (3)   ±  —  Vd'  —  x^. 


INCKEMENTS. 

9.  If  the  independent  variable  be  made  to  change  from  one 
value  to  another,  the  quantity  by  wliich  it  is  changed  is  called 
its  Increment;  and  this  increment  is  positive  or  negative  accord- 
ing as  the  variable  is  increasing  or  decreasing. 

When  the  independent  variable  receives  an  increment,  the 
corresponding  change  in  the  value  of  any  function  of  it  is  the 
increment  of  this  function,  and  is  found  by  subtracting  the  old 
from  the  new  value  of  the  function.  Hence,  if  the  function  is 
increasing,  its  increment  is  jDOsitive;  and  if  decreasing,  its  in- 
crement is  negative. 

The  increment  of  a  variable  is  denoted  by  writing  the  letter 
/I  {delta)  before  it.  Thus,  ^x  does  not  mean  A  times  x,  but 
"the  increment  of  x,"  and  is  so  read.  Similarly,  Au,  A{x^),  and 
z/(ic^  +  '^^)  denote  the  increments  of  u,  x^  and  x''  +  ^x. 

In  this  book  h  will  generally  be  used  instead  of  Ax,  as  it  is 
more  convenient ;  but  it  should  always  be  remembered  that 
h  =  Ax. 

Increments  and  the  method  of  finding  them  are  illustrated 
algebraically  and  graphically  in  the  solution  of  the  following 
examples. 


6 


DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 


Illustrations.     1.  If  x  is  iucreased  by  li,  what  will  be  the 

increment  of  the  function  it  =  ex  ? 

A  p  p  The  value  of  u  (=  ex)  unay  be 

represented  graphically  by  the 
area  of  the  rectangle  OB  PA, 
whose  base  OB  =  x,  and  whose 
altitude  OA  =  c. 


Au 


,x 


B       h 


Fig.  1. 


u  =  ca;  =  OBPA. 


(1) 


When  a;  is  increased  by  BC  {=  7i),  the  new  value  of  u  will  be 
c{x  +  /i),  or  area  of  OOP' A.  Hence,  denoting  the  increment  of 
w  by  Jii,  we  have 

w  +  ^tt  =  cx  +  ch  =  OOP' A (2) 

Subtracting  (1)  from  (2),  member  from  member,  we  have 

Aii  =  cli  =  BGP'P (3) 

That  is,  the  increment  of  ex  is  c  times  the  increment  of  x. 

2.  By  how  much  will  u  =  x^  be  increased  when  x  is  increased 
by /i? 

The  value  oi  u  {=  x'')  may  be  represented 
graphically  by  the  area  of  the  triangle  OBP 
whose  base  OB  =  x  and  whose  altitude  BP 


'X.     Hence 


x'  =  OBP. 


(1) 


When  OB  (=  x)  is  increased  hj BC{=  h) 
the  new  values  of  u,  x^  and  OBP  will  be,  re- 
spectively, u  +  z/w,  {x  +  liY  and  OOP',  thus 
changing  (1)  into 

u  +  Au  =  x'  -^  2xn  +  7i'  =  OOP'.       .     .     . 
Subtracting  (1)  from  (2),  member  from  member,  we  have 
J2i  =  2xh  +  h'  =  BOP'P; 


or 


Jw  =  2a7i  +  Ji'  =^BPx  h  +  PDP'. 


(2) 


(3) 


FUNDAMENTAL  PRINCIPLES. 

3.  Find  the  increment  of 

y  =  x^  —  ^x  -\-  6.       .    . 


(1) 


X  B     C 

Fig.  3. 


We  may  regard  ?/  =  a;^  —  4a;  +  5  as 
the  equation  of  a  curve  APP';  then,  P     ^ 
being  any  point  of  the  cnrve,  x  =  OB 
and  ?/  =  BP. 

When  OB   {=  x)  is    increased    by 
BC  {=  h),  the  new  values  of  y,  x"  —  4a;+ 5 
and  BP  will  be,  respectively,  y  +  /iy, 
{x  +liy  -4:{x  +  li)  +  b,  and  CP',  thus 
changing  (1)  into 

y  +  Ay  =  x"  -\-  2xh  +  h""  -  4a;  -  4/i  +  5  =  CP\     .     (2) 

Subtracting  (1)  from  (2),  we  have 

/ly  =  {2z  -  4)7i  +  h'  =  CP'  ~  BP  =  DP'.    .     .     (3) 

CoE.  I.  Evidently  DP'  will  be  +  or  —  according  as  CP'  is 
greater  or  less  than  BP;  that  is,  in  general,  Ay  will  be  positive 
or  negative  according  as  y  is  increasing  or  decreasing. 

10.  General  Formula.     To  find  the  increment  of 

u=m (1) 

■  Increasing  x  by  Ti,  and  denoting  the  corresponding  increment 
of  w  by  Au,  we  have 

u  +  Ati  =  f{x  +  h) (2) 

(2)-(l),  Au=f{x-\-h)-f{x). 

EXAMPLES. 

1.  Find  the  increment  of  ii  =  nx^,  or  the  areaof  a  concentric 
ring  whose  width  is  li  and  whose  inner  radius  is  x. 

Au  -  7r{2x  +  Ji)h. 

2.  Find  the'increment  of  the  cube  u  =  x\ 

Au  -  dx'h  +  Sxh"  +  h\ 


8  DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

3.  Find  the  increment  oif{x)  =  a:'  —  7a;  +  9. 

f{x  +  h)  -fix)  =  {'5x'  -7)h  +  Sxh'  +  h\ 

4.  Given /(.r)  =  x' +  2x'  +  9;  show  that 

f{x  +  h)  -  fix)  =  i?,x'  +  ^x)li  +  (3a;  +  2)A'  +  h\ 

5.  Given /(a-)  =  Vx;  prove  that 

■va;  +  /i  +  y  a; 

6.  Given  w  =  — ;  prove  that  /^w  = j-— — ^. 

VARIATION. 

11.  Proportional  Variation.  One  quantity  is  said  to  vary- 
proportionally  with,  or  to  vary  as,  another  when  the  ratio  of 
the  one  to  the  other  remains  constant. 

The  sign  of  variation  is  a.  Thus,  y  varies  as  x  is  written 
y  <xx. 

Illustrations.  1.  The  cost  per  yard  of  cloth  remaining  the 
same,  the  entire  cost  iy)  varies  as  the  quantity  (.r).     That  is, 

y  ^^ (1) 

2.  The  space  (s)  described  by  a  body  moving  with  a  uniform 
velocity  iv)  varies  as  the  time  it).      Or 

s  cxt (2) 

3.  The  area  {u)  of  a  rectangle  having  a  constant  altitude 
[a)  varies  as  the  base  (a;).     Or 

w  a  a- (3) 

12.  Principles.  I.  //  one  quantity  varies  as  another,  one 
of  til  em  is  a  constant  multiple  of  the  other. 

Let  y  '^  ^, 

then  —  =  m,  a  constant; 

X 

hence,  y  =  mx. 


FUNDAMENTAL  PRINCIPLES.  9 

CoE.  I.  The  variations  (1),  (2)  and  (3),  Art.  11,  may  be 
written,  respectively, 

y  =  rax,  where  m  is  the  jirice  per  yard  of  cloth; 
s  =  mi,  where  m  is  the  velocity  of  the  body; 
u  =  onx,  where  m  is  the  altitude  of  the  rectangle. 

13.  II.  If  one  variable  is  equal  to  a  constant  multiple  of 
anotlier,  the  first  varies  as  the  second. 

Let  y  =  mx, 

where  m  is  a  constant.     Then 

y 

—  =  m,    or     y  a  x. 

X  -^ 

14.  III.  If  each  of  two  quantities  {u  and  v)  varies  as  a 
third  (x),  the  first  quantity  (u)  varies  as  the  second  (v). 

Since  u  a  x,  u  =  /nx; (1) 

and  since  v  oc  x,  v  =  nx (2) 

(1)  H-  (2),  -  =  — ,     or     u  =  — y. 

That  is,  u  cc  v. 

15.  IV.  The  prod^ict  of  two  variables  does  not  vary  pro- 
portionally wifh  either  of  the  variables. 

Let  us  suppose  yx  a  or, 

then  yx  =  mx,     or     y  =  7n, 

which  is  contrary  to  the  hypothesis,  since  m  is  a  constant. 

16.  Disproportional  Variation.  When  one  quantity  does 
not  vary  as  another,  the  variation  is  said  to  be  disjDroportional. 
Hence,  one  quantity  varies  disproportionally  with  another  when 
the  ratio  of  the  one  to  the  other  does  not  remain  constant. 

Thus,  x^  varies  disproportionally  with  x,  since  x^  -^  x  {=  x) 
is  not  a  constant. 


10  DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

17.  Principles.  I.  The  jjrodud  of  two  varialUs  varies  dis- 
proidortioncdly  tuith  eitlier  of  the  variables  (Art.  15). 

18.  II.  Tlie  nth  looioer  of  a  variable,  n  having  any  value  ex-, 
cept  -\-  1,  varies  disjjroportionally  with  the  variable. 

For,  a;"  -^  x  {~  a;""')  is  a  constant  only  when  n  =  +  1. 

19.  If  mji  vanishes  luith  h,  mji^  varies  disiwoiiortionally 
with  h. 

Let  us  suppose  mji^  a  7^; 

then  mji"^  =  mh,     or     7nji  ~  m; 

this  being  true  for  all  values  of  h,  must  be  true  when  7i  =  0, 
hence,  since  mji  A'anishes  witli  h,  the  value  of  the  constant  m  is  0. 
Therefore  mjr  varies  disprojiortionally  with  h  if  m^  is  of  such  a 
cliaracter  that  mji  vanishes  with  7^.* 

20.  Composition  of  Increments.  In  general  the  increment 
of  any  function  of  a  single  variable  is  composed  of  two  j)arts, 
one  of  which  changes  proportionally,  and  the  other  disj^ropor- 
tionally,  with  the  increment  of  the  variable. 

Thus,  let  y  =/(a')  represent  any  function  of  x,  h  any  variable 
increment  of  x  estimated  from  any  particular  value  of  x,  and 
Ay{—-f[x-\-  h)  —f{x)  )  the  corresponding  increment  of  y;  then 

/jy  =  mji  +  mJi", (1) 

where  m^  is  constant  with  respect  to  h,  and  m^  is  of  such  a 
character  that  mji  vanishes  with  h  (Art.  19). 

The  proof  of  this  property  of  increments  will  be  given  in 

*  To  say  that  m2h  vanishes  with  h  is  equivalent  to  saying  that  m^  does 

not  involve  any  negative  power  of  h,  such  as  — ,  —  ,  etc. ;  for  if  it  did,  the 

value  of  m-Ji,  when  It  =  0,  would  be  finite  or  infinite.  In  general,  mih- 
varies  disproportionally  with  Ji  whatever  may  be  the  character  of  ;ho  ,  but 
the  only  cases  we  shall  have  occasion  to  consider  are  those  in  which  mnh 
vani.shes  with  Ji. 


FUNDAMENTAL  PRINCIPLES.  11 

Ar^,.  24;  we  give  here  two  examples  in  illustration  of  tliis  im- 
portant proposition. 

1.  Take  the  increment  of  w  =  x^,  which  we  have  found  to  be 
Au  =  2xJi  +  h\ 

By  reference  to  Fig.  2,  it  will  be  seen  that  BCDP  =  2xJi  and 
PDF'  =  h\  Eegarding  BF  (=  2x)  as  constant,  and  BC  (=  h) 
as  variable,  the  first  part  of  Au,  BCDF,  varies  proportionally 
with  h,  and  the  second  part,  FDF',  disproportionally  with  h. 

By  comparison  with  (1),  m^  =  2x  and  m^  =  1. 

2.  Take  the  function  ii,  =  x^  —  lx-\-  9,  the  increment  of 
which  we  have  found  to  be 

Au  =  {?>x^  -  7)A  +  (B.r  +  Ji)?i\ 

Here,  regarding  x  as  constant,  the  j)art  (3a;^  —  7)h  varies  as 
//-  (Art.  13),  and  the  other  part,  (3:^;  +  h)h-,  changes  disj)ropor- 
tionally  with  h  (Art.  19).  Comparing  this  increment  with  (1), 
we  have 

m^  =  dx'^  —  7     and     771^  =  3x  -\-  It. 

THEORY  OF  LIMITS. 

21.  The  principles  of  limits,  in  addition  to  other  merits, 
aiford  an  admirable  method  and  means  of  finding  the  propor- 
tional increments  of  related  variables.  For  convenience  of  refer- 
ence, and  in  order  that  the  method  of  proportional  changes  and 
that  of  limits  may  be  made  to  throw  light  upon  each  other,  we 
give  here  a  statement  of  such  of  the  princijoles  of  limits  as  we 
shall  have  occasion  to  employ. 

The  Limit  of  a  variable  is  a  constant  which  the  variable  ap- 
proaches, and  from  which  it  can  be  made  to  differ  by  less  than 
any  quantity  which  may  be  assigned,  but  which,  on  the  other 
hand,  it  can  never  actually  reach. 

Thus,  if  the  number  of  sides  of  a  regular  polygon  inscribed 
in,  or  circumscribed  about,  a  circle  be  indefinitely  increased,  the 
area  of  the  circle  will  be  the  limit  of  the  area  of  either  polygon, 
and  the  circumference  will  be  the  limit  of  the  perimeter  of 
either. 


12  DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

Again,  let  i?  =  1  +  |  +  i  +  -|  +  etc. 

By  increasing  tlie  nnmber  of  terms  of  this  series,  the  value 
of  .s  approaches  2  indefinitely,  but  can  never  reach  it;  therefore 
2  is  tlie  limit  of  s. 

22.  Principles.  I.  The  difference  'between  a  variable  and 
its  liuiit  is  a  variable  'whose  limit  is  0. 

23.  II.  If  two  variables  are  continually  equal,  and  each 
a'p'proaches  a  limit,  their  limits  are  equal. 

For,  let  X  =  Y  be  the  variables  and  A  and  B  their  respec- 
tive limits,  and  suppose  X  -\-  x  =^  A  and  Y  -\-  y  =^  B,  then 
X  —  y  ~  A  —  B;  but  the  limits  of  x  and  y  are  0  (Art.  22), 
therefore,  at  the  limit,  ^  —  ^  =  0,  or  ^  =  _5. 

24.  Proof  of  the  Formula  Jy  =  mji  +  mji\     .     (Art.  20) 
Let  y  =f{x)  be  a  continuous  function  such  that 

^  0,  /(^  +  /0  -fix) 
/ix  h 

approaches  a  definite  limit  (say  m^)  as  It  approaches  zero,*  then 

Ay 
the  value  of  -y-  must  be  of  the  form  m^  +  '>nji>  where  mji  is  a 

quantity  whose  limit  is  0  or  one  which  vanishes  with  h. 

Ay 
That  is,     -—  =  m^  +  mJi".     .:  Ay  =  mji  -\-  mji\ 

where  m^  is  evidently  independent  of  h. 

Geometrical  iLLUSTRATiON.f  Let  y  —  /(a;),  as  defined 
above,  be  the  equation  of  a  curve  APm,  where  x—  OB  and 
y  =  BP.  Let  TPt  be  a  tangent  to  the  curve  at  P.  When  x 
is  increased  hj  BC  =  h,  we  have  Ay  =  DF\  Draw  the 
secant  SPP',  and  we  have 

Au       DP' 

-£  =  TD=  '-  """P (') 

*  See  Appendix,  Ai.      f  For  another  illustration  see  Appendix,  Aj. 


FUNDAMENTAL  PRINCIPLES. 


13 


ISTow  taking  the  limits  of  tlie  two  members  of  (1),  remember- 
ing that   as  h  ajoproaches  0,  P' 
approaches    P,    and    XSP    ap- 
proaches XTP   as   its  limit,  we 
have 

m^=  tan  XTP, 

which  is  a  definite  quantity  de- 
pendent on  a  particular  value 
{%'  =  OB)  of  X,  and  independent 
of  h. 

Again,  since  m^  =  tan  XTP,  mji  —  Dt,  and  since 

/iij  =  DP'  =  mji  +  mji% 

m^i"^  =  tP' ,  which  is  a  quantity  that  vanishes  with  //. 


DIFFEREXTIALS   AND   ACCELERATIONS. 

25.  The  Diflferential  of  a  function  is  that  part  of  its  incre- 
ment which  varies  proportionally  with  the  increment  of  the 
independent  variable,  and  the  Acceleration  is  that  part  which 
varies  disprojiortionally  with  the  increment  of  that  variable. 

Thus,  Art.  20,  (1),  the  differential  of  y  is  mji,  and  the  accel- 
eration of  y  is  mji'-. 

The  differential  of  a  quantity  is  denoted  by  writing  the  letter 
d  before  it.  Thus,  dy  is  not  d  X  y,  but  the  differential  of  y, 
and  is  so  read.  The  differentials  of  functions  like  x^,  x^  -f  '^■'^j 
and  l^l  \-  x",  are  denoted  by  d{Q:^),  d{x''  ~\-  7x),  and  d{Vl  +  re"). 

Sitnilarly,  the  acceleration  of  a  function  will  often  be  denoted 
by  writing  the  letter  a  before  it;  as,  ay,  which  is  read,  "the  ac- 
celeration of  y." 

CoE.  I.  The  increment  of  a  function  is  equal  to  the  sum  of 
its  diffei'ential  and  its  acceleration.     Thus, 


^1/  =  dy  +  (^y- 


14  DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

The  acceleration  may  be  positive  or  negative. 

Cor.  II.  When  x  and  y  are  indejaendent  variables.  Ax  =  clx, 
and  Ay  —  cly. 

The  independent  variable  may  be  snpposed  to  change  in  any 
manner  whatever;  its  increments  are  arbitrary,  and  are  themselves 
independent  variables  dependent  not  even  on  the  value  of  the 
independent  variable  itself,  while  the  increments  of  the  depend- 
ent variable  depend  on  both  the  independent  variable  and  its 
increments.  This  arbitrary  character  of  the  independent  varia- 
ble leaves  us  free  to  make  the  most  convenient  supposition  with 
reference  to  the  manner  of  its  variation,  which  is  that  this 
variation  is  uniform,  or  that  its  increments  have  no  acceleration 
but  are  differentials. 

Cor.  III.   The  differential  of  a  function  at  any  value  is  what 

its  succeeding  increment  would  be  if  at  that  value  its  change 

became  proportional  to  that  of  the  increment  of  the  independent 

All 
variable.     That  is,  when  y  =  f{x),  (1)  the  limit  of  -j-,  as  h 

cly 

approaches   0,  is  1,  and  (2)  dy  <x  h.     For  example,  in  Fig.  4, 

DP' 

we  have  (1)  the  limit  of  —j=n-,  as  h  approaches  0,  is  1,  and  (2) 

J-JZ 

Dt  a  PD. 

Since  the  differential  of  the  function  varies  as  the  increment 
of  the  independent  variable,  the  former  will  vary  uniformly  when 
the  latter  does. 

Cor.  IV.  The  differential  of  a  function  is  positive  or  negative 
according  as  the  function  is  increasing  or  decreasing. 

Cor.  V.  In  the  increment  Ay  =  7nji  -\-  mji",  dy  =  mji  = 
m^dx,  and  ay  =  mji\  Hence  in  Fig.  4,  since  Ay  =  DP',  and 
dy  =  D.f,  we  have  ay  =  tP'. 

dv 
26.  In   the   equation   dy  =  m^dx,  m^  or  -—  is   called  the 

Derivative  or  Differential  Coefficient  of  y  with  respect  to  x,  and 
is  equal  to  tan  XTP,  Fig.  4. 


27. 


FUNDAMENTAL  PRINCIPLES.  15 

Ay 
Since  the  limit  of  -^_,  as  Ax  approaches  0,  =  m^,  Art. 


24.  and  since  -—  =  m, ,  we  have 
ax         ^ 


limit 
Ax^O 


■Ay- 
Ax 


=  -^      or     Fio-  4 


which  is  read,  "  the  limit  of  -p,  as  z/a;  approaches  0,  is  eqnal 

to  -T^."     We  are  to  understand  by  this  that  the  ratio  of  the  pro- 

loortioiral  variations  of  y  and  x  can  be  found  b}^  taking  the  limit 

of  -r-.    The  student  should  note  that  the  limits  of  Aii  and  Ax  are 
Ax  "^ 

not  dy  and  dx;  but  the  limit  of  -p  is  equal  to  -j-,  because  each 

is  equal  to  m^,  just  as  ^  is  equal  to  |f,  because  each  is  equal 
tof. 

28.  In  finding  the  differential  of  a  function  it  is  not  neces- 
sary to  find  the  entire  increment;  only  the  part,  or  parts,  involv- 
ing the  first  power  of  h  or  dx,  will  be  sufficient,  for  the  terms 
which  involve  the  higher  powers  of  1i  form  the  acceleration  of 
the  function. 

As  an  example  let  us  find  the  differential  of 

u  =  x'  —  5x'  +  dx (1) 

Increasing  x  by  7i,  we  have 

u  +  Au  =  {x  +  hy  -  5 (a-  +  7iy  +  3{x  +  7i) 

=  x'  +  4:x'h  +  etc.  -  5{x'  +  3x'7i  +  etc.  -f  S{x  +  7i),     (2) 

where  the  omitted  terms  contain  higher  powers  of  7i. 

(2)  — (1),  f/w  =  {4:X^  —  15a;'  -f  3)dx,  Ans. 


16 


DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 


29.  Cor.  I.  If  u  and  v  are  functions  of  x,  Avhen  x  is  increased 
by  h  the  proportional  increments  of  u  and  v  are  du  and  dv,  and 
these  are  the  parts  of  Au  and  Av  which  involve  the  first  power 
of  A. 


DIFFERENTIALS  OF  GEOMETRIC  FUNCTIONS. 

30.  Differentiation  is  the  operation  of  finding  the  differen- 
tial of  a  fnnction  in  terms  of  the  differential  of  its  variable.  The 
process  consists  in  finding  the  increment  of  the  function  and 
removing  from  it  the  acceleration,  or  in  determining  what  the 
entire  increment  would  be  if  it  varied  as  the  increment  of  the 
variable. 

The  following  important  formulas  are  deduced  at  this  time 
more  especially  for  the  purpose  of  illustrating  the  preceding- 
principles. 


31.  Differential  of  Plane  Areas 
in  Rectangular  Co-ordinates.  Let 
APP'  be  any  plane  curve,  OB  =  x', 
BP  =  y,  and  area  of  OB  PA  =  ic; 
it  is  required  to  find  the  differential 
(du)  of  u. 

When  X  is  increased  hj  BG  {=  h), 
we  have 


B        C 


Fig.  5. 


Ate  =  BCP'P  =  BCDP  +  PDP', 

which  corresponds  in  form  with  Ay  =  mji  +  m.Jt",  Art.  24,  in 
which  mji  =  BCDP,  mJi'  =  PDP' ,  m,  =  BP  =  y,  and  m.p 
vanishes  with  Ti. 

Since  the  initial  side  of  Au  is  BP  (=  y),  the  rectangle  BCDP 
(=  yli)  is  what  Au  would  be  had  it  varied  as  li;  hence  the  area 
of  the  rectangle  is  the  differential  of  w. 


•.  du  =  y7i     or    ydx. 


FUNDAMENTAL  PRINCIPLES.  17 

Or  thus:  since  the  limit  of  BCP'P  ^  BCDP,  as  BC  ap- 
proaches 0,  is  1,  and  since  BCDP  a  BC,  BCDP  (=  ydx)  is  the 
differential  of  OB  PA  (=  ti). 

Method  bij  Limits.  The  increment  BCP'P  is  >  BCDP  and 
<BCP'Q\  that  is. 

All 
yAx<  Au<  {y  +  Ay)Ax;     .:  y<  ^<  y  +  Ay. 


All 
As  Ax  approaches  0,  the  limit  of  Ay  is  0,  and  that  of  —r-  is 

equal  to  ^- ;  hence  we  hav^e 

^  ax 


du  ,1    ,  •     du  ,  , 

y<-j-<y',     that  IS,  -j--  =  y,     or    dn  ~  ydx. 


Hence,  the  required  differential  is  equal  to  the  value  of  the 
ordinate  exjjressed  in  terms  of  x,  multiplied  iy  the  differential 
of  the  abscissa. 

Cor,  I.  In  Fig.  5,  au,  the  acceleration  of  u,  is  the  area  of 
PDP\ 

32.  Differentials  of  Solids  of  Revolution  in  Rectangular 
Co-ordinates.  In  Fig.  5,  let  v  eqnal  the  volume  of  the  solid 
generated  by  the  revolution  of  OB  PA  about  OJ^  as  an  axis;  it 
is  required  to  find  the  differential  of  v. 

When  X  is  increased  by  h  the  corresponding  increment  of  v  is 
the  volume  of  the  solid  generated  by  revolving  BCP'P  about 
BC  as  an  axis;  that  is, 

Av  =  vol,  generated  by  BCP'P 

=  vol,  gen'd  by  BCDP  +  vol.  gen'd  by  PDP' 
=  TTy'h  +  vol.  gen'd  by  PDP'. 


18  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

Since  the  initial  base  of  Av  is  the  circle  whose  radius  is  y, 
the  cylinder  generated  by  BCDP  {=  Tty^li)  is  what  the  incre- 
ment Av  would  be  had  it  varied  as  li. 

dv  =  Tiy^li    or     ny^dx. 

Metliod  liy  Limits. 

Vol.  gen'd  by  BCDP  <  Av  <  vol.  gen'd  by  BCP'Q, 

or  Tty-Ax  <  Av  <  7t{y  +  AyyAx. 

Passing  to  limits,  as  in  i^revious  examples,  we  obtain 
dv  =  Tty'^dx. 

Hence,  the  required  differential  is  rt  times  the  square  of  the 
value  of  the  ordinate  expressed  in  terms  of  x,  multijjlied  by  the 
differential  of  the  abscissa. 

CoE.  I.  In  Fig.  5,  av,  the  acceleration,  is  the  volume  gener- 
ated by  PDP\ 

33.  Differential  of  the  Arc  of  a  Plane  Curve  in  Rectangu- 
lar Co-ordinates.  In  Fig.  4,  let  s  =  the  length  of  the  arc  AP, 
then  PP'  =  As.  Since  Dt  is  what  Ay  would  be  had  it  varied 
as  h,  Pt  is  what  As  would  be  had  it  varied  ash;  hence,  Df  =  dy 
and  Pt  =  ds;  and  since  Pt  =  VPD''  +  Dt',  we  have 


ds  =  Vdx-  +  dy-,     or     f  y  1  +  ^)dx. 

Hence,  the  required  differential  is  the  square  root  of  the  sum 
of  one  and  the  square  of  the  value  of  the  derivative  of  y  with 
resjject  to  x,  exjjressed  in  terms  of  x,  multiplied  by  the  differential 
ofx. 


FUNBAMEFTAL  PRINCIPLES. 


19 


Cor.  I.  TJie  limit  of  the  ratio  of  an  arc  of  any  plane  curve 
to  its  chord  is  unity. 
For  (Fig.  4), 

As_ 
arc  PP'  As  Ax 


chord  PP'        »/^  _|_  -jy- 
the  limit  of  which  is  evidently  unity. 


^^^m 


34.  Differential  of  Surfaces  of  Revolution  in  Rectangular 
Co-ordinates.  Let  s  =  the  length  of  the  arc  AP,  and  8  =  the 
area  of  the  surface  generated  y 
by  the  revolution  of  ^P  about 
OX  as  an  axis,  then  AS  ^  the 
area  of  the  surface  generated  by 
the  revolution  of  PP' {=  As)  ^ 
about  BC  {=h). 

At  P  and  P'  draw  PG  and    q 
P'Q    each  equal  to  PP'  and 
parallel  to  OX;  the  areas  of  the 

surfaces  generated  by  revolving  PG  and  P'Q  about  OX  are 
2nyAs  and  27t{y -{- Ay)As,  ^i\i\.  evidently  z/>S'  lies  between  the 
former  and  the  latter.     That  is. 


27tyAs  <  AS  <  "litiij  +  Ay)As. 

27ry  <  ^  <  2;r(y  +  Ay). 

As  h  approaches  0,  the  limit  of  Ay  is  0,  and  the  limit  of 
is  equal  to  -y- ;  hence  we  have 

,7  o 


A8^ 
As 


or    dS  =  ^Tcy  Vdx"  +  %^ 


20  DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

Hence,  the  required  differential  is  the  product  of  the  circum- 
ference of  a  circle  lohose  radins  is  y,  iy  the  differenticd  of  the 
arc  of  the  generating  curve. 

35.  Differential   of  Plane   Areas   in   Polar  Co-ordinates. 
Let  APP'  be  any  plane  curve,  and  let  0  be  the  pole,  OP  (=  r) 
^p         the   radins   vector,    and    put   B  -- 
XOP  and  u  —  the  area  of  OA  P ;  it 
is  required  to  find  the  differential 
of  u. 

When  6  is  increased  by  the 
angle  POP'  (=  ZIO),  u  will  be 
increased  by  the  area  of  POP' 
(=  Au). 

From  0  as  a  centre  with  the  radius  Oa  (=1)  describe  the 
arc  be  (=  Ad  or  dO),  and  with  the  radius  OP  (=  r)  describe  the 
arc  PD  (=  rdO). 

Since  OP  is  the  initial  side  of  Au,  the  area  of  the  sector  POD 
is  what  Au  would  be  had  it  varied  as  Ad. 

du  =  ^OP  X  PD  =  ir'dd. 

POP' 

Or  thus:  since  the  limit  of  -pTyrf,  as  be  approaches  0,  is  1, 

and  since  POD  a  be,  POD  (=  ^r^dS)  is  the   differential  of 
GAP  (=  u). 

OoR.  I.  The  acceleration  of  u  is  the  area  of  PDP\ 


CHAPTEE  II. 
ELEMENTARY  DIFFERENTIATION  AND  INTEGRATION. 

DIFFERENTIATION. 

36.  Every  function  may  be  differentiated  by  the  principles 
.    heretofore  established,  but  in  practice  it  is  better  to  use  rules, 

which  we  now  proceed  to  deduce. 

37.  To  differentiate  a  constant.  Since  a  constant  has  no 
increment,  its  differential  is  0.     That  is, 

dc  =  0,    and    dC  =  0. 

Hence,  the  differential  of  a  condant  is  0. 

38.  To  differentiate  the  product  of  a  constant  by  a 
variable. 

Let  u  —  cv, (1) 

where  c  is  a  constant  and  y  is  a  function  of  x. 

Let  h  represent  any  variable  increment  of  x  estimated  from 
any  particular  value  of  x,  and  dti  and  dv  the  corresponding  pro- 
portional increments  of  u  and  v,  Art.  29;  then 

u  +  du  =  c{v  +  dv)  =  cv  -\-  cdv (2) 

(2)  —  (1),  du  =  cdv (3) 

.*.  Rule. — Multiply  the  constant  hy  the  differential  of  the 
variable. 

39.  To  differentiate  the  product  of  two  variables. 

Let  u  =  vy, (1) 

where  v  and  y  are  functions  of  x. 

21 


22  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

Let  x'  represent  any  particular  value  of  x,  and  u',  v' ,  y'  the 
corresponding  values  of  u,  v,  and  y;  then 

n'  =  v'f (1) 

Let  h  represent  any  variable  increment  of  x  estimated  from 
x',  and  chi,  dv,  and  dy  the  corresponding  proportional  incre- 
ments of  V,  V,  and  y  (Art.  29);  then 

u'  -{-  du  =  {v'  +  dv){y'  +  dy)  —  {dv){dy) 

=  v'2/  +  y'dv  4-  v'dy (2) 

The  term  {dv){dy)  is  eliminated,  or  dropped  from  the  second 
side,  since  it  varies  disproportionally  with  h  (Art.  17)  and  is 
therefore  a  part  of  the  acceleration  of  u.  Subtracting  (1)  from 
(2)  we  have 

dtc  =  y'dv  -\-  v'dy. 

Since  y'  and  v'  are  any  corresponding  values  of  y  and  v,  we 
have 

du  =  d{vy)  =  vdy  -f-  ydv. 

.'.  Rule. — Multiply  the  first  by  the  differential  of  the  second, 
and  the  second  by  the  differe7itial  of  the  first,  and  add  the  two 
products. 

40.  To  differentiate  the  product  of  any  niimber  of 
variables.     Let  u  =  vyz,  where  v,  y,  and  z  are  functions  of  x. 

Assume  w  =  yz, 

therefore  u  ■=  vio\ 

then  du  =  {w)dv  -\-  v{dw), 

also  dw  =  zdy  +  ydz; 

therefore  d2i  —  {yz)dv  +  v{zdy  +  ydz) 

=  yzdv  +  vzdy  +  vydz. 

In  a  similar  manner  it  may  be  shown  that 

d{viuyz)  =  vimjdz  -f  vwzdy  +  vyzdio  -j-  ivyzdv. 


ELEMENTAR  Y  DIFFERENTIA  TION  AND  INTEGRA  TION.  23 

.*.  EuLE. — Take  the  sum  of  the  products  obtained  hy  multi- 
plying the  diferential  of  each  hy  all  the  other  variables. 

41.  To  differentiate  a  fraction. 

Let  u  =  —,  where  v  and  y  are  functions  of  x. 

y  ^ 

Since  w  =  — ,  we  have  iiy  =  v. 

y 

Differentiating,        udy  +  ydu  =  dv.  (Art.  39) 

,  ,        dv dy 

TTTi                      1          dv  —  udii                y   ^ 
Whence,  du  =  ■ = ~ 

y  y 

_  ydv  —  vdy 

/.  EuLE. — Multiply  the  denominator  by  the  differential  of 
the  numerator  and  from  the  product  subtract  the  numerator 
multiplied  by  the  differential  of  the  denominator,  and  divide  the 
result  by  the  square  of  the  denominator. 

42.  Cor.  I.  The  differential  of  a  fraction  whose  numerator 
is  a  constant  is  minus  the  numerator  into  the  differential  of  the 
denominator  divided  by  the  square  of  the  denominator. 

For,  Art.  41,  d[-]  =  ^  ^  ~  ^  -^  =  —  ^-^,  since  dc  =  0. 

^yi  y  y 

43.  To  differentiate  a  variable  having  a  constant  ex- 
ponent. 

Let  u  =  v" ,  where  v  is  any  function  of  x,  and  n  is  any  con- 
stant integer  or  fraction. 

I.  When  the  exponent  is  a  positive  integer. 

Since  v^  =  v  .v  .v .  .  .  to  n  factors, 

d{v'')  =  v""Vv  +  v^'-^dv  .  .  .ion  terms       (Art.  40) 

=  nv^~  dv. 


24  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

II.  When  the  exponent  is  a  positive  fraction. 

a 

Let  u  =  v" ,  then         u"  =  v": 
Differentiating,  cu''~'^du  =  av'^'hlv. 


( 
Substituting  for  «,     cv 

iTJ- 

•1 
d7i 

=  «V" 

hlv. 

Eeducing, 

a  - 
CV 

a 

du 

=  av"'' 

-'dv. 

a 
a-- 

Dividing  by  cv        , 

du 

c 

«     c 

=    -V 
C 

dv. 

III.  When  the  exponent 

is  negative, 

Let  u  =  -y"™,  then 

u  — 

1 

Differentiating,  Art.  43, 

dv  = 

=    — 

iv 

-    

mv"'"" 

'dv. 

Hence,  for  all  the  cases,  we  have  this 

Rule. — Tahe  the  product  of  the  exponent,  the  variable  with 
its  exponent  diminished  by  1,  and  the  differential  of  the  variable. 

Thus:  d{x'')  =  bx'dx,        d{ir')  =  -  7u-'d2i, 

d{v^)  =  Iv^dv,        d{z-^)  =  -  iz-^dz, 

d{r)  =  hj-'dy  =  ^. 

The  same  rule  holds  when  the  exponent  n  is  irrational  or 
imaginary. 

44.  Cor.  I.  The  differential  of  the  square  root  of  a  variable 
is  the  differential  of  the  variable  divided  by  twice  the  radical. 


ELEMENTABT  DIFFERENTIATION  AND  INTEGRATION.   25 

For;  d{  4/y )  =  div")  =  iv-^dv  =  — ^—, 

2  Vv 

45.  To  differentiate  the  algebraic  sum  of  several 
variables. 

Let  u  =  V  +  y  —  z, (1) 

where  v,  y,  and  z  are  functions  of  x. 

Let  h  represent  any  variable  increment  of  x  estimated  from 
any  particular  value  of  x,  and  du,  dv,  dy,  and  dz  the  correspond- 
ing proportional  increments  of  u,  v,  y,  and  z;  then 

u  -\-  du  =  V  -\-  dv  -[-  y  -\-  dy  —  {z -{-  dz).      .     .     (2) 

(2)  —  (1),  du  =  dv  +  dy  —  dz. 

.'.  EuLE. — Take  the  algelraic  sum  of  their  differentials. 

EXAMPLES. 
Differentiate  the  following: 

1.  y  =  x'  +  5.^'  -  3x  +  7. 

dy  =  d{x')  +  d{bx')  -  d{3x)  +  d{7)  (Art.  45) 

=  d{x')  +  5d{x')  -  Sd{x)  +  d{7)  (Art.  38) 

=  3x'dx  -f  lOxdx  —  3dx  +  0  (Arts.  37,  43) 

=  {3x'  +  10.r  -  3)dx,  Ans. 

2.  y  =  x'  4-  5a;  +  3.  dy  =  {2x  +  5)dx. 

5.  y  =x'  -  4.r'  -  3x\  dy  =  {4x'  -  123;'  -  (yx)dx. 
A.  y  =  3x'  -  Ix'  -  9.  dy  =  [Vox'  -  etG.)dx. 

f>.  y  =  x'  +  2x'  -  7a;  +  10.  dy  =  {Q>x'  +  lO.c^  -f  etc.)f7a;. 

6.  y  =  a;'  +  x-\  dy  =  {2x  -  2a;-2)ffe. 

7.  2/  =  «;-»  +  x^'  dy={-  3a;-*  +  ^x~")dx. 

8.  y  =  x^  —  «a;-*+  b.  dy  =  {^x^  +  4:ax-^)dx. 

9.  y  =  Qx^  —  4a;^  +  2a;l  %  =  (21a;^  —  etc.)f/a;. 

10.  y  =  3a;~*  —  Qx~^  +5.  (?y  =  (—  .r"^  +  Ax~^)dx. 

11.  y  =  «a;™  —  &a;~".  (^?y  =  {max^~'^  +  b?ix~'^'^)dx. 

-^       2^3a;^|/^ 


26  DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 


13.  y  =  \/x  +  -.     {-x'  +  o:-\)     dy  =  {^x  ^  —  x-'-)dx, 

14.  y  =  1+ ,7-     {.^ax-^-Vhxr\)    dy  ==  (-  ^  -^j'^^- 


15.  y'^-=  4:ax. 

dy  =  ~  dx. 

IG.  y'-^x'  ■=  R\ 

dy  = dx. 

y 

17.  aSf  +  &-^x-  =  «^^*\ 

dy  = ^dx. 

ay 

Find  the  following: 

18.  d{x'+  l)(x'  +  2.r). 

{x'  +  l)^(a;'  +  2a;)  +  {x'  +  2a;)rf(a;^  +  1)       (Art.  39) 
=  {x'  +  1)(2^-  +  2)dx+  (:«'  +  2x)3.Tdr 
=  (2a;'  +  2:c'  +  2a;  +  2)^.t;  +  {tx'  +  4.«')c?.c 
=  (4.r='  +  6a;'  +  ^x  ^  1)dx,  Ans. 

19.  fZ(a;  -  1) (.*;'■'  +  a;  +  1).  3a.''^Za-. 

20.  d\ax'y^).  4:ax'y'dx  +  Sax*y''dy. 

21.  (Z(:c)0/^  +  1).  (2/^  +  l)dx  +  2T^y^?/. 

22.  d{x'  -  l)(a;^  +  1).  {Qx'  -  4:x'  +  2a;)f/a-. 

23.  di6x^y').  dx'^yhlx -\- 2xy~^dy, 

24.  ^[a;-''(l  +  x-%  (-  2.^"^  -  5x-')dx. 

25.  (^(1  +  a;)(a;  +  a;')a;. 

26.  ^(1  +  2a;'0(l  +  4.r'). 

27.  ^(a;  +  l)(a;'  —  2;"^  +  a;  —  1).        4:x'dx. 

28.  ^(a;'  +  a)(3a;'  +  b).  {I5x'  +  3bx'  +  6aa;)da;. 

29.  ^(12a;^2/^  +  15aJ6-).  ISx^y^dx  +  16a;^3/^t?y. 

Differentiate  the  following: 

30.  «  =  ]+^. 

1  —  x 


ELEMENT AR T  DIFFERENTIA  TION  AND  INTEGRA  TION.   27 

(1  —  X) 

2dx 

-J ,  Alls. 


(I  -  xY 


adx 


31.  u  =  .  du  = 

X  X' 

1  +  a;  ,         (1  -  2x  -  x')dx 

'''-  ''  =  r+^^-  ^^"  =  —jr+^^y—' 

x^  -,  Sx^dx 

33.  M  =  T-, ^-  «« 


a-'  x'     ■  ,  2.aY/.T 

x-  —  I    X  —  1  (x-  —  ly 

x' 


35.  u  =  — ,- 


a'  —  X 


.2' 


2a;^  -  3  ,         8x'  +  6.r  +  12  , 

3C.  u  = ■ — 5.  du  =  —- — , — JTT— f^^- 


4.r  +  a;*'  ~      (4a;  +  .^^)^ 

3 
{a  +  6a;)' 

a;° 
(1  +  ^>r  "'^^  ~  (i  +  -'^')= 


37.  u  = 

{a 

x'  ,         Zx'  +  x\. 


39.  21  =  Vx^  —  3.r  +  5. 

2  i^a;'  -  3.r  +  5        2  ^ x^  -  3a;  +  5 

40.  w  =  y  3.r  —  5.  «z^  =  -— :=z=.— . 

4/3a;^  —  5 

41.  w  =  l^ia;  +  |/9^'.  f??^  =         .-^fl^^i'. 

2  l^a; 

2  +  3.T 


42    ?(  =  .T'/l  +  a;.  f??^  =  — ; '—^dx. 

2Vl-\-x 

X  -.  dx 

43.  u  =  — .  a2i 


Vl  -x-"  (1  -  a;=) 

1  +  a; 
\  —  X 


i/l  +  x  '       -,  dx 

44.  76  -  y  — -^ — .  du  — 


(1  -  a-)  1^1  -  X' 


28  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

45.  li  =  {x'  -  Zx"  +  4.x  -  by. 

du  =  5(a;=  -  3a;'  +  Ax  -  byd{x'  -  3:r'  +  4x  -  5)       (Art.  43) 
=  b{x^  —  3.r'  +  Ax  -  5)'(3.^•'  -  Qx -{-  4)dx. 

46.  w  =  (x'  -  Ix  +  5)'.  du=?>{x'-lx-^by{^x'-l)dx. 

47.  u  =  (i^;'  +  5a;  -  9)^  du  =  ^{x''+bx-^f{;Zx-tb)dx. 

'ddx 


48.  ti  =  (24/a;4-  3)-^  </w  = 


Vx{2Vx  +  3)*' 


,^             ,     a;     \                                  -           wza;      dx 
49.  ?<  = .  du  = 


l-xj   '  (i_a;f+^* 

a  —  Sx) 

2V^r^ 

Sx'  +  2  dij  2 


KA  /     1     \a/ 7         («  ~  ''ix)dx 

50.  w  =  (ff  +  ;i-)r  a  —  x.  du  =  -^ 

2l/«  -  a; 

51.  2/  = 


x{x'^\f  dx           x\x'^\f 

V{x  +  «?  f/y       (:?;  -  2a)^a;  +  a 

52.  y  =  — ^-=k^z^.  T^  =  ^ '- — ~ — . 

\x  —  a  ^-^              {x  —  ay 


SLOPE  OF  CURVES. 

46.  Direction.  The  direction  of  a  straight  line  is  deter- 
mined by  its  angle  of  direction,  which  is  the  angle  formed  by 
the  axis  of  x  and  the  line. 

The  Slope  of  a  line  is  the  tangent  of  its  angle  of  direction. 
Thus,  in  Fig.  4,  the  angle  of  direction  of  the  line  TP  is 

XTP,  and  the  slope  of  the  line  is  tan  XTP  =  y^  =  j-' 

47.  The  direction  of  a  curve  at  any  point  is  the  same  as 
that  of  a  tangent  to  the  curve  at  that  point. 

For,  at  the  point  P  (Fig.  4),  the  deviation  of  the  secant 
SPP'  from  the  curve  PP',  arising  from  the  former's  cutting 
the  latter,  diminishes  indefinitely  as  P'  approaches  P  ;  and 
since  the  tangent  TPt,  which  touches  the  curve  at  P,  is  the 
limiting  position  of  the  secant,  the  tangent  ,has  the  same  direc- 
tion as  the  curve  at  the  point  P  or  {x,  y). 


ELEMENT AB T  DIFFERENTIA TION  AND  INTEGRA TION.   29 

48.  CoE.  I.  The  slope  of  a  curve  at  any  point  is  the  slope 
of  its  tangent  at  that  point.     Therefore  the  slope  of  a  curve  at 

the  point  {x,  y)  is  - -.     The  differential  of  the  arc  of  a  curve  at 

any  point  is  a  straight  line  laid  off  on  the  tangent  to  the  curve 
at  that  point,  Art.  33. 

49.  The  angle  of  direction  of  a  curve  at  the  point  {x,  y)  is 
usually  denoted  by  0;  that  is,  0  =  XTP,  Fig.  4. 

,       dy        .      ,       dij  ,  ^       dx 

.:  tan  0  =  ^,     sm  0  =  —,    and     cos  0  =  -r-, 
dx  da  ds 

where  ds  =  the  differential  of  the  arc  of  the  curve.  Art.  33. 


EXAMPLES. 

1.  The  equation  of  a  curve  is  y  =  x^  —  ^x;  find  the  slope  of 
the  curve  at  the  point  {x,y). 

Differentiating  the  equation  and  dividing  by  dx,  we  have 

4^  =  3x'  -  2,  Ans. 
dx 

2.  In  the  same  curve  find  the  slope  at  the  point  where  x  =  1. 

dx 

3.  Find  the  slope  of  the  parabola  ?/"  =  ^x  at  the  point  where 

x  =  ^.  ^  =  1 

dx       4 

4.  In  the  same  parabola  find  the  point  where  the  curve  makes 
an  angle  of  45°  with  the  axis  of  x. 

Since  tan  45°  =  1,  we  make  -^  =  —  =  1;   hence  2y  —  9, 

?/2       9 
or  ?/  =  4i,  and  re  =  ^-  =  -  =  2^. 

5.  In  the  circle  y"^  -\-  x^  ■=  R'^,  find  the  point  where  the  slope 
of  the  curve  is  —  f.  AVhere  y  =  ^R,  and  x  =  |i?. 


30  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

6.  In  the  same  circle  find  the  point  where  the  curve  is  par- 
allel with  the  line  whose  equation  is  5y  -j-  12:?;  =  60. 

Where  x  =  j^E. 

7.  At  what  angles  does  the  parabola  y^  =  6x  cut  the  circle 
y'  +  x'  =  16  ? 

Find  their  slopes  at  their  points  of  intersection;  then  find 
the  angles  between  the  lines  having  these  slopes.  Thus:  solving 
the  two  equations,  we  find  (for  one  of  the  points  of  intersection 
of   the  two  curves)   x  =  2  and  y  =  2  i/3.     The   slope   of  the 

parabola  at  this  point  is  —  (^=  ^  V'd),  and    that  of  the  circle 

is ^  (=  —  ^  4/3).     Therefore  the  tangent   of   the   required 

angle  is 

11/3-^1  V3      ^  I  1/3  ^  5 


V3  =  2.88  +. 


8.  At  what  angles  does  the  line  3y  —  2.i;  —  8  =  0  cut  the 
parabola  y'^  =  8x  ?  Tan"^  .2  and  tan~^  .125. 

9.  The  equation  of  a  curve  is  y  =  x^  —  9x^  -\-  24a;  —  11. 
(1)  Find  the  slope  of  the  curve  at  the  point  where  x  =  3.  (2) 
Find  the  values  of  x  at  the  points  where  the  slope  of  the  curve 
is  45.  (3)  Find  the  values  of  x  at  the  points  where  the  curve  is 
parallel  with  the  axis  of  x. 

(1)  —  3;  (2)  a;  =  7  and  -  1;  (3)  x  =  2  and  -'. 

10.  Find  the  point  Avhere  the  curve  y  =  x''  —  I'x  -\-  3  is  par- 
allel to  the  line  y  ^  5x  -\-  2.  Where  a;  =  6. 

11.  Find  the  point  where  the  parabola  y^  =  'iax  is  parallel 
with  the  circle  y"^  —  2Rx  —  a;^  Where  x  =  R  —  2a. 

12.  Show  that  the  ellipse  7-5  +  "3-  =  1  intersects  the  hyper- 

lo        8 

bola  x"^  =  ?/^  +  5  at  right  angles. 

13.  At  what  angles  does  the  circle  x^  -\-  y'^  =  Sax  intersect 

x^ 

the  cissoid  ?/^  =  ^ ? 

■^        2a  —  X 

At  the  origin,  90°;  at  the  other  two  points,  45°. 


ELEMENTAB  T  DIFFERENTIA  TION  AND  INTEGRA  TION.   31 


INTEGRATION. 

50.  Integration  is  the  inverse  of  differentiation,  Thus, 
while  differentiation  is  the  process  of  deriving  the  differential  of 
a  function  from  the  function,  integration  is  the  process  of  de- 
riving the  function  from  its  differential.  The' function  is  called 
the  integral  of  the  differential.  Thus,  the  differential  of  x^ 
being  dx'dx,  x^  is  the  integral  of  ox^dx. 

The  Sign  of  integration  is  /  ;  thus  /  'dx'dx  indicates  the 
operation  of  integrating  3x^dx,  therefore  /  'ix'^dx  —  x\  which  is 
read  "  the  integral  of  3x\lx  is  x\"    The  two  signs  d  and    /  annul 

each  other;*  f  d{x')  =  x'  and  dj  {3x'dx)  =  ?,x\lx. 

51.  Dependent  Integration.  When  the  process  of  inte- 
grating depends  on  reversing  the  corresponding  process  of  dif- 
ferentiating, as  is  usually  the  case,  it  is  called  dependent  inte- 
gration. This  process  will  now  be  employed  in  establishing 
rules  for  integrating  elementary  differentials. 

52.  To  integrate  0. 

Since  dC  =  0,  Art.  37,  we  have    A  =  C. 

Therefore,  as  0  may  be  added  to  any  differential  without 
changing  its  value,  the  general  form  of  its  integral  will  not  be 
complete  without  an  indeterminate  constant  terra.  This  con- 
stant term,  as  will  be  seen,  may  be  eliminated,  or  determined 
from  the  data  of  any  particular  problem. 

53.  To  integrate  the  product  of  a  differential  and  a 
constant. 

Since  d{cv)  =  cdv,  Art.  38,  we  have    /  cdv  =  c  j  dv  =  cv  -\-  C. 


*  To  be  exact,  dff{x)dx  =j\x)dx,  but  fdj[x)  =f{x)  +  G. 


32  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

.".  Rule. — Integrate  tJie  differential,  and  multiply  the  result 
hy  the  constant. 

54.  To  integrate  v''dv,  where  n  has  any  positive  or  nega- 
tive, integral  or  fractional  value  except  —  1. 

Since  d  — —7    =  v'Ulv,  Art.  43,  we  have 
\n  +  1/ 


,,"+1 


/  v\lv  =  -^,  +  C. 
J  n  -\-l 

.'.  Rule. — Remove  dv,  increase  the  exponent  ly  1,  and  divide 
the  result  by  the  new  exponent. 

Thus,  Jx\lx  =  \x''  +  C',  fifdy  =  |2/'  +  C; 

Cv-hlv  =  -  iv-^  +  C;      fdu-idu  =  6ui  +  0. 

55.  Cor.  I.  The  above  rule  applies  also  to  /  {v^dv,  where  v 
is  any  fnuction  of  a  variable. 

Thus,    f{x'  -  3^  +  5y[2x  -  3]dx  =  i{x^  -  ^x  +  5)'  +  O. 

In  this  example  v  =  x^  —  3x  -\-  5,  and  dv  =  {2x  —  3)dx;  that 
is,  the  rule  applies  whenever  the  factor  without  the  (  ),  viz., 
[2a:;  —  3]dx,  is  the  differential  of  the  quantity  within  the  (  ). 

56.  Cor.  II.     /  v"do  =  —  /  {v)"cdv;  introducing  a  constant 

thus  is  sometimes  necessary  to  render  the  quantity  without  the 
(  )  the  differential  of  the  one  within.     Thus, 

fiix  +  2a;')^(l  +  x)dx  =  ij^i^x  +  2x'y{4:  +  4x)dx 
^-3^{4x  +  2x^y^+C. 

57.  CoE.  III.  Any  differential  of  the  form  (a  +  Ix'^fx^^dx, 
where  n  =  m,  -\-  1,  can  be  integrated  by  the  above  rule;  thus, 


f{a  -f  Ix'^x'^-Hx  =  \f{a  +  ix'Yndx''-' 


dx 


ELEMENTARY  DIFFERENTIATION  AND  INTEGRATION   33 

7ih^   ^  /     \      1         /  w6( jt)  +  1) 

Thus,     fx  |/3  +  bx'  dx  =  ^f{d -\-  5x')Kxdx  =  ^^(3  +  5a;')'  +  0. 

/(5TW"  "^^^  ^  '  •^•')"*^'^-'^-  =  -  6V(5  +  ^0;=)-^  +  O. 

The  rule,  Art.  54,  does  not  hold  for  n  =  —  1,  for  the  reason 
that  d{v^)  is  not  v'^dv,  but  0.  The  formula  for  this  case  will  be 
derived  subsequently.  Art.  180,  formula  3. 

58.  To  integrate  the  algebraic  sum  of  two  or  more  dif- 
ferentials. 

Since  d{u  -{-  v  —  z)  =  du  +  dv  —  dz,  (Art.  45) 

we  have       /  {du  +  dv  —  dz)  =    I  du  +   /  dv  —    I  dz 

■=u  -\-  V  —  z-{-  C. 
.'.  EuLE. — Tal-e  the  algebraic  sum  of  their  integrals. 

Thus,    y^(3a;'  -  2;r  +  b)dx  =J'6x^dx  -  Jlxdx  -^Jbdx 

=  x'  -x"  +  bx-\-  a 

EXAMPLES. 
Integrate  the  following: 
1.  dy  =  {Qx'  -  4.T  +  D)dx. 

y  =  jQx'^dx  -  J  ^xdx  4-  Jhdx  (Art.  58) 

=  6  Jx\lx  -  4  Jxdx  +  5  fdx  (Art.  53) 

=  6-|  -4-|- +5-.'K+ C'  (Art.  54) 

=  2.t'  -  2.1;'  +  5.^  +  C,  Ans. 
3.  dy  -  (^ix'  -  5x*)dx.  y  =  x'  -  x'  -}-  C. 

3.  dy  =  {5x  —  3x~  )dx.  y  =  fa;  +  a;~  +  C*. 


34  DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

4.  dij  =  (3a;*  +  2x''')dx.  y  -  2z^  —  x'^  +  C. 

5.  dy  =  {x^  -\-  x^)dx.  y  =  %x^  -\-  fa;^  +  C. 

clx  1  2 

6.  dy  =  ^-^  or  x~^dx.  y  =  ^x^  +  C. 

Vx 

7.  dy  =  -^  ov  x~  dx.  y  =  —  ■Ja;"''  +  C. 

8.  (Zi/  =  (ax''  -\ =\dx.  y  =  iax'  +  i/^+  C. 

\  2  Vx  I 

10.  cZ_y  =  (1  +  a;*)la;'fZ.r. 

y=f\{l  +  xyAxhlx  =  1^(1  -f  xyd{l  +  a:^)     (Art.  56) 

=  tV(1  +  --c')'  +  0,  Ans.     (Art.  55) 

11.  dy  =  (1  +  a;)Wa;.  y  =  i(i  _^  a;)^  4_  (7. 

12.  f/y  =  (1  +  xfdx.  y  =  1(1  +  a;)^  +  C'. 

13.  dy  =  (1  -  a;)"'^:r.  ?/  =  (l  _  x)~^  +  C. 

14.  fZ?/  =  (rt  +  a;'j*a;^a;.  ?/  =  -!(«  +  a;^)^  +  C. 

15.  fZ^  =  («  -  a;^)"V^.r.  y  =  -  ^a  ~  x')^  +  (7. 

16.  dy^{b  +  x'^f-x~\lx.  ?/  =  _  i(J  +  a:"')'  +  0. 

Find  the  following: 

17.  J{x'  -  ^x-')dx.        ,  Ix'  +  x~^  +  a 


4/1 


a; 


zi.  f{^=M^^^=]dx.  V2x' -ex +  5  +  a 

•^    \  i'2x'  -  6.T  +  5/ 


ELEMENT AB  Y  DIFFERENTIA  TION  AND  INTEGRA  TION.   35 

33    f   ^^^    .  — - —  +  a 

■  ^  (1  -  a;^')'  4/1  -  x' 

25.   r.J^i_  jW^b  +  a 

^     ■t/4a;^^  -  5 

26.  r_J^ziML_.  iy'3.T^-(i.r  +  7+a 
^    4/32;'  -  6a;  +  7 

27.  ^(1  -  a;=)Ur.  a;  -  a;^  +  l--^'  -  t--^'  +  ^. 

28.  /Y^+^VrZ..  _«_l+o,,  +  '^+a 
'y    \       X      J  X  o 

29.  f(^--±^±^yx.  2  ^x{a  +  i&.r  +  ic.^=)+  a 

30.  /  /a:(«^  -  x^Ydx.  2a;^(^  _  ^f^^  ^  -  ^)  +  (7. 

The  following  differentials  may  be  reduced  to  the  form  of 
(a  +  Ix^Y x'^-'^dx,  and  then  integrated  by  Art.  57, 


r    x-^dx      _    r      x-^dx  \/a;  + «;'        „ 

^^-  J    ^a'  +  x'  ~'^    Va'x-'  +  1  •  a^ 

Multiply  the  binomial  under  the  radical  sign  by  x'^,  and  the 
numerator  of  the  fraction  by  x~^. 

^^      r  j/x'  -  a'  dx  i^"  -  «')'  _^  Q 
^^      r  V2ax  -  x'  dx  {lax  -  x'f 

34      f       ^^       .  J=+^. 

J  {ce-\.x'f  a'Va'  +  x' 

35.  r_ii^^— ,  —=!_-  + a 

^    (2aa;  —  a;')*  o  r  ^a:^  -  x' 


36  DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 


PROBLEMS. 

59.  The  Problem  of  Integration  is  the  inverse  of  that  of 
differentiation ;  if  the  hitter  is  "  given  a  curve  to  find  its  slope," 
the  former  is  "given  the  slope  of  a  curve  to  find  the  curve"; 
if  the  latter  is  "  given  a  function  to  find  its  rate  of  change," 
the  former  is  "given  the  rate  of  change  of  a  function  to  find 
the  function,"  Since  the  general  problem  of  differentiation  is 
"given  a  function  and  its  variable  to  find  their  proportional 
changes,"  that  of  integration  is  "given  the  proportional  changes 
of  a  function  and  its  variable  to  find  the  function." 

60.  Definite  Value  of  the  Constant  C.  To  complete  each 
integral  as  determined  by  the  preceding  rules,  we  have  added 
a  constant  quantity  C.  While  the  value  of  this  constant  is  un- 
known, it  is  said  to  be  indefi?iite  ;  but  it  becomes  definite  when 
its  value  is  assigned  or  determined  by  the  conditions  of  the 
problem  under  consideration.  The  signification  of  C  and  the 
manner  of  determining  its  definite  value  are  illustrated  in  the 
following  examples. 

EXAMPLES. 

1.  Required  the  equation  of  the  curve  whose  slope  at  the 
the  point  {x,  y)  is  4;c'  —  2a;  +  3. 

By  Art.  48,  the  slope  of  a  curve  at  {x,  y)  is  -—. 

%  =  4:x'  -  2a;  +  3, 
ax 

or  dy  —  (4a;'  —  2x-\-  3)dx. 

Integrating,  y  =  x*  —  x'  -{-  3x  -]-  C, 

which  is  the  indefinite  integral  or  equation  required. 

To  determine  the  value  of  C  we  may  make  x  =  0,  which 
gives  y^  =  C,  where,  y„  indicates  what  y  becomes  when  x  —  0, 
and  is  therefore  the  distance  from  the  origin  to  where  the  curve 
cuts  the  axis  of  y. 


ELEMENTARY  DIFFERENTIATION  AND  INTEGRATION.  37 

Therefore  the  equation  may  be  written 

y  =x'  —x"  +  ^x  +  y, , 

which  becomes  definite  when  y^  is  known. 

When  we  know  any  corresponding  values  of  an  integral  and 
its  variable,  (7  can  b>  determined.  Thus,  if  it  is  given  that  the 
last  curve  passes  through  the  point  x'  =  2,  ?/'  =  10,  then  when 
a;  =  2,  ^  =  10,  and  we  have 

10  =  2*  -  2'  +  3  2  +  (7,     or     (7  =  -  8. 

2.  What  is  the  equation  of  a  curve  which  passes  through  the 
origin,  and  whose  slope  at  the  point  [x,  y)  is  {a  +  a;)'  ? 

Here  ,^  =  (a  -(-  ^;)^     or     dy  =  {a  -\-  xydx. 
Integrating,  y  z=  ^a -\-  xY  +  C. 

Since  the  curve  passes  through  the  origin,  y^  =  0;  hence, 
making  x  =  0,  we  have 

0  =  ia'+C;     .:   0  =  -  ia% 

and  the  required  equation  is 

y  =  i{a  +  xY  -  ia\ 

3.  Required  the  equation  of  a  curve  which  passes  through 
the  point  {x'  z=  3,  y'  =  11),  and  whose  slope  at  {x,  y)  is  3.^;'  — 
lO.r  +  1.  y  —  x^  —  5x'  +  a;  +  26. 

4.  The  differential  of  an  integral  is  (3  +  x')^xdx,  and  the 
integral  is  0  when  x  =  0;  required  the  value  of  0. 

c=  -  1/3. 

5.  The  differential  of  a  function  is  (1  +  fx^dx,  and  the 
function  is  0  when  x  —  0;  find  the  function. 

A(l  +  i-^)'  -  A- 

6.  A  function  and  its  variable  vanish  simultaneously,  and 

their  proportional  changes  are  as  x\a''  +  x'')~^  to  1;  find  the 
function.  |(a'  +  x^)^  —  §«. 

Take  dx  for  the  increment  of  x,  then  a;°(rt"  +  x^)~^dx  will  be 
the  proportional  increment  of  the  function. 


38 


DIFFERENTIAL  AND  INTEGRAL   CALCULUS.. 


7.  Find   the   area   of   a   plane   curve    whose   differential   is 
V4:ax  dx,  and  whose  value  =  0  when  x  =  Q.  fa;  V^ax. 

8.  "When  x  is  increased  by  dx  the  proportional  increment  of 
the  arc  of  a  certain  curve  is  (9  +  x")  x^dx;  find  the  length  of 
the  arc,  supposing  it  and  x  to  start  from  the  same  point. 

i(9  +  xf  -  4i. 

9.  The  area  of  a  surface  of  revolution  is  increased  propor- 
tionally by  7r(l  -{-  X  —  .T^)~^(l  —  2x)dx  when  x  is  increased  by 
dx;  find  the  area  of  the  surface,  if  it  =  0  when  x  =  0. 

[{1  +  x-x'f  -  l]27r. 

10.  The  differential  of  the  volume  of  a  solid  of  revolution  is 
7r{2Rx  —  x'^)dx;  find  the  volume  supposing  it  and  x  to  vanish 
simultaneously.  Tt^Rx^  —  ^x^\ 

61.  Applications  to  Geometry.  The  last  four  problems 
indicate  the  possibility  of  finding  the  length  and  area  of  plane 
curves  and  the  area  of  surfaces  and  volume  of  solids  of  revolu- 
tion when  their  differentials  are  known.  Now  Arts.  31,  32,  33, 
34  enable  us  to  find  the  differentials  when  the  equation  of  the 
curve  is  given. 

62.  Areas  of  Curves. 

11.  Find  the  area  of  OBPA,i\\Q  equation  of  the  curve  APE 

being  y  =  x""  —  Sx  -\-  15, 
where  x  =■  OB  and  y  = 
BP. 

By  Art.  31,   the    differ- 
ential  of    OB  PA  {=  u)   is 
-X  ydx,  and  in  the  jDresent  ex- 
j,jg  g  ample  y  =  x'  -  8x  +  15. 

Hence,  dit  =  ydx  —  (x^  —  8x  -\-  15)dx; 

.-.  u  =f{:x"  -8x  +  lb)dx  =  ^t'  -  4x''  +  Ibx  +  C. 
Evidently  the  area  ^<  =  0  when  x  =  0,  hence  C  =  0. 


ELEMENTAR  Y  DIFFERENTIA  TION  AND  INTEGRA  TION.   39 

63.  Definite  Integrals.  If,  iu  any  indefiuite  integral,  two 
different  values  of  the  variable  be  substituted,  aiid  the  one  result 
subtracted  from  the  other,  C  will  be  eliminated,  and  the  result 
is  called  a  definite  integral. 

Thus,  Fig.  8,  if  Oa  =  1  and  Ob  =  2,  to  obtain  the  area  of  the 

re' 
section  alccl  we  substitute  2  for  x  in  '- —  4.r^  +  15:c  +  C  and 

o 

get  16f  +  C'  (=  area  of  OicA),  and  then  substitute  1  for  x  and 

obtain  11-g-  +  6'  (=  area  of  OaclA);  subtract  the  latter  from  the 

former  and  we  have  5^  (=  area  of  abed). 

The  following  is  the  notation  by  which  these  operations  are 

indicated : 


/ 


\x'  -  8x+16)dx=  [^  —  4.t--'  +  15a-  +  C 


1U  =  5*. 


In  general,    /     is  the  symbol  of  a  definite  integral,  and  indi- 

cates  (1)  that  the  differential  following  it  is  to  be  integrated; 
(2)  that  b  and'a  are  to  be  substituted  successively  for  the  variable 
in  the  indefinite  integral;  and  (3)  that  the  second  of  these  re- 
sults is  to  be  subtracted  from  the  first.  Tlie  operation  is  called 
integrating  between  the  limits  a  and  b. 

13.  In  the  last  example  find  the  area  of  EFG. 

Since  the  roots  of  x'^  —  Sx  +  15  =  0  are  3  and  5,  OE  =  3 
and  OG  =:  5.     Hence  we  have 

/  ydx  =     —  -  4rc'  +  15x-\-  a\    =  -  1^-  =  area  of  EFG. 

The  result  is  negative  because  the  area  lies  below  the  axis  of  x. 
13.  In  the  same  example  find  the  area  of  OEA. 


I 


du  =  18. 

0 


14.  Find  the  general  area  of  the  curve  y  —  S.r^  —  2.t  +  3 
estimated  from  the  origin.  /''^    7, .  _    a  _    »  4_  q 

15.  Find  the  areas  of   the  positive  and  negative  surfaces 
enclosed  by  the  curve  y  =  x'^  —  x  and  the  axis  of  x. 

J_^ydx  —  \;    J  ydx=—i. 


40 


DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 


16.  Find   the   entire  area   of  ^'^  =:  (1  +  a;')a;'  between  the 
origin  and  the  point  whose  abscissa  is  x.  |(1  +  ^')'  —  !• 

17.  Find  the  positive  area  of  the  parabola  y"^  =  4ax.         %xy. 

18.  Find   the   area  of  y^  —  x\a^  —  x^)  between  the  limits 
.T  =  0  and  a;  =  a.  J-^^a  _  ^s^*^,^^  ^  ^^.^ 

64.  Lengths  of  Curves. 

19.  Find  the  length  of  the  arc  OP, 
the  equation  of  the  curve  OP  being 
y^  =  ax^. 

By  Art.  33,  the   differential   of  OP 

(=  5)  is  yi  +  (yj'f/.-^.     To  apply  this 

to  the  present   curve,  we  differentiate 
y"^  =  ax^,  and  obtain  2ydy  =  dax^dx. 
dy  _  Sax''      idyV  _  9a V  _  ^ax 
^®"^®'  dx~^^'     [dxl  ~"^~T'' 


Fig.  9. 


ds 


30.  The  differential  of  the  equation  of  a  certain  curve   is 
1  dx;  find  the  length  of  the  curve,  beginning  at  the 


dy  =  Vx' 
origin. 

Here 


dy 
dx 


=  Vx'  -  1; 


-\-\  =  x' 


=  ¥ 


21.  Find  the  length  of  the  arc  of  a  curve  whose  equation  is 


—    1  2.^2 


X'  — 


y  z=  |(ir  —  1)%  measured  from  the  point  where  x  =  1 

23.  Find  the  length   of  a  curve  the  differential  of  whose 

equation  is  dy  =  \^x''  -|-  2xdx,  beginning  at  the  origin. 

ix'  +  X. 

x^        1 
23.  Find  the  length  of  the  curve  y  =  ^  4-  -,  between  the 


limits  (1),  X  =  2,  X  =  3;  (3),  x  =  a,  x  =  b. 


(1)  I;  (2) 


b'  -a' 
13 


ab 


ELEMENTARY  DIFFERENTIA  TION  AND  INTEGRA  TION.  41 


24.  Find  the  general  length  of  the  curve  y  =  \1  —  '^]^x, 
estimated  from  the  origin,  /         x\   ,— 


65.  Areas  of  Surfaces  of  Bevolution. 

25.  AP  is  the  arc  of  the  circle  ?/''  =  ^^—.^';    f^ 
find  the  area  of  the  zone  generated  by  re- 
volving AP  about  the  axis  OX. 

By  Art.  34,  dS='Z7nj\/l  +  i^j)'dx;  in 

this  example,  since  y^  =  R^  —  x'^,  ~  =  —  '-. 
^  ^  ax  y 

Hence, 


27r?/l/  1  +  -„  dx  =  27rEdx. 

y 


dS  =  27rvY 
.-.  S  =  iTtRj   dx  =  27tRx. 


26.  In   ex.  25   find  the  area  of  the  zone  generated  by  the 
arc  PP',  supposing  OB  =  a  and  OB  =  i.     rgr'  —  27rR(b  —  a) 

'  a 

27.  In  ex.  25,  find  the  surface  of  the  generated  sphere. 

28.  Find  the  surface  of  the  cone  which  is  generated    by 
revolving  the  line  y  =  mx  about  the  axis  OX. 

{Sf=  TTyVx'  +  y'. 

29.  Find    the  surface   of    the   paraboloid,   the    generating 
curve  being  the  parabola  y^  =  4:ax. 

[Sf=i7rVa[{a  +  x)^-{a)^l 

30.  Find   the  surface   generated   by  the  revolution  of  the 
curve  y  =  ax^  about  the  x  axis.  x         [(1  +  9rtV)'  —  11 

•^"^^0  =  "^  27^  • 


42  DIFFERENTIA f.  AND  INTEGRAL   CALGULU8. 

66.  Volumes  of  Solids  of  Revolution. 

31.  Fiud  the  volume  generated  by  revolving  OBPA  about 

the  axis  OX,  the  equation  of 
the  curve  AF  being  y'  =  '6x' 
—  36.T  +  105. 

By  Art.  32,  dv  =  ny'^dx;  in 
this  example  2/^= 3a;^  — 36a:-)- 105, 
-^   hence 

dv  =  7r(3a''  —  36x  -\-  105)dx. 

Z6x  +  105)^a;  =  7r{x'  -  18:<;=  -f  lOS.r). 

32.  In  the  same  example  find  the  volume  of  the  solid 
generated  by  the  revolution  (1)  of  OEA;  (2)  of  FGF. 

(1)  [vf=  2007r;    (2)  [v]' =  -  47r. 

0  5 

33.  Find   the  volume   of  the  cone  which  is    generated  by 

revolving  about  x  the  triangle  whose  base  is  x  and  whose  altitude 

is  y  {=  mx).  P"",    5,,,         7r??/V  Jx\ 

^  v  =  7tj^    {m^x^)dx  =  ^-=  rry^[-j. 

34.  Find  the  volume  of  the  solid  which  is  generated  by 
revolving  ?/  =  a;'  —  4  about  OX. 

V  =  7tf{x'  -  8a;'  +  U)dx  =  7i[\x'  -  fa;'  +  16:c]  -f-  C. 

35.  In  ex.  34,  find  the  volume  generated  by  the  negative 
area  of  the  given  curve.  r^-]+^  _  3^2  jj; 

3G.  Find  the  volume  of  the  spherical  segments  generated 
(1)  by  OBPA  and  (2)  by  BEP'P,  Fig.  10;  also  find  (3)  the 
volume  of  the  sphere  (see  Ex.  26). 

(1)  [i'f  =  7t{R^a  -  i«');  (2)  [v]  =  7t[R\h  -a)-  i(5'-a')]; 

0  a 

(3)  [vi^=  i7rR\ 

37.  Find  th-e  volume  of  a  prolate  spheroid,  the  generatrix 
being  the  ellipse  d'y''  =  a'6^  —  b''x\  r^'l^"=  ^TraF. 


CHAPTER  III. 

SUCCESSIVE    DIFFERENTIALS    AND    RATE    OF 
CHANGE. 

SUCCESSIVE  DIFFERENTIALS. 

67.  The  differential  of  any  function,  as  w  =/(.?;),  denoted  by 
du,  is  the  fiist  differential.  The  differential  of  the  first  differ- 
ential, viz.,  d{dn),  denoted  by  d'u,  is  the  second  differ e7itial. 
The  differential  of  the  second  differential,  viz.,  d{dhi),  denoted 
by  d^ii,i&  the  tliird  diff^erential ;  and  in  general  the  differential 
of  d"~'^u,  denoted  by  d''u,  is  the  7it]i  differential. 

du,  d^ti,  d^u,  etc.,  are  the  Successive  Differentials  of  n. 

68.  If  X  is  the  independent  variable,  its  differential  [dx)  is 
altogether  independent  of  x, — the  increment  given  x  at  any  in- 
stant being  entirely  independent  of  the  value  which  x  may  have 
at  that  instant.  It  at  once  follows,  therefore,  that  the  differential 
of  dx  with  respect  to  x,  like  the  differential  of  any  other  variable 
which  is  independent  of  x,  is  0.     Hence  d{dx)  =  d  '\v  =  0. 

For  example,  take  the  function  u  =  x^;  then 

(1)  du    =  3x^dx', 

(2)  d'u  =  d{3xyix  +  Zx"-d{dx)  =  Qxdx'; 

(3)  d'u  =  d{Qx)dx'  +  6xd{dx')  =  6dx'; 

(4)  d'u  =  d{6)dx'    +  6d{dx')    =  0. 

Therefore,  in  finding  the  successive  differentials  of  a  func- 
tion, we  treat  the  differential  of  the  independent  variable  as  a 
constant.  See  Appendix,  A^,. 

The  student  should  observe  the  difference  between  expres- 
sions like  d-y,  di/\  and  d{i/-);  dh/  is  the  second  differential  of  y, 
dif  is  the  square  of  the  differential  of  t/,  and  d(y'')  is  the  differ- 
ential of  //'',  or  2ydi/, 

43 


44  DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

If  u  =  f{x),  the  successive  derivatives  or  differential  coef- 

flcients  of  u  with  respect  to  x  are  -^,  ^^,  -j-^,  etc. 

The  successive  derivatives  ot  f{x)  are  also  denoted  by /'(a;), 
f"{x),f"\x),...r{x). 

Therefore,  when  u  =  f{x),  we  have 

EXAMPLES. 

1.  Find  the  successive  differentials  of  y  =  x*. 

dy   =  ^xhlx. 

d'^y  =  d{Ax^dx)     =  ^dxd{x'')      =  \2x^dx^. 
d'y  =  d{l2x\lx')  =  12dx'd{x')  =  24:xdx\ 
d*y  =  d{24:xdx')  =  2^dxhl{x)   =  24:dx\ 
d'y  =  d{24tdx*)     =  0. 

2.  Find  the  successive  derivatives  of  x^  —  4x''  +  Sa;  —  5, 
Let  f{x)  =  x'  -  4.x'  +  ^x-b; 

then     f'{x)     =  j{x'  -  4:X-  +  3.t  -  5)  =  Zx'  -  8x  +  d; 
f"{^x)   =  y  i^x'  -8x^3)  =  Qx  -  8; 

/"(^')=  1^(6.^ -8)  =  6; 

/^  (x)  =  0. 

3.  Find  the  successive  differentials  of  7i  =  2x^  —  '7x*. 

du  =  (Gx'  -  14:X)dx;  d\i  =  {\2x  -  14)ffe';  d\i  =  \2dx\ 

4.  u  r=:  x"  —  Zx^  -\-  5x. 

5.  u  =  x'^. 

du  =  —  3x~h2x;  d'^u  =  \2x~^dx'';  d^u  =  —  QOx'^dx^;  etc. 

6.  n  =  x~^  —  x~^  -\-  5. 

dii  =  —  {2x~^  —  x'''^)dx;  d'^u  =  {Qx'^  —  2x-^)dx;  etc. 

7.  u  =  {x  +  1)1 

du  =  i{x  +l)~\/a;;  d'u  =  -  ^{x  +  lyhlx';  etc. 

8.  If  fix)  =  -4-^,  show  that /'''(a;)  ='-  ,    ^  ,.,. 


SUCCESSIVE  DIFFEBENTIALS  AND  BATE  OF  CHANGE.   45 


*  o* 


9.  If  fix)  =  ^^-,  show  that/'^(l)  =  ^i^ . 

10.  Uf{x)  =  1^,  show  that/-(l)  =^. 

11.  If  f(x)  =  ^i|,  show  that/'"(0)  =  -  |. 

69.  Leibnitz's  Theorem.  To  find  the  successive  differentials 
of  the  product  of  two  variables. 

Let  u  and  v  be  functions  of  x;  then 

d{nv)  =  iidv  +  v(?w (1) 

Differentiating  (1),  regarding  u,  v,  du,  dv,  as  functions  of  x, 
we  have 

d^{uv)  =  icd^v  +  f??<f7y  4-  dvdu  -\-  vd^u, 

or  d'^{uv)  =  nd^v  +  2dudv  +  t'f^'w (2) 

Differentiating  (2),  we  get 

d^{uv)  =  nd°v  -\-  Mud'v  +  ^d''udv  +  vd^u, 

from  which  we  see  that  the  law  of  the  coefficients  is  evidently 
the  same  as  in  the  binomial  formula. 

RATE    OF    CHANGE. 

70.  Uniform  Change.  When  a  variable  changes  uniformly 
it  experiences  equal  changes  in  equal  intervals  of  time,  whatever 
the  magnitude  of  these  intervals. 

Thus,  we  may  suppose  the  line  ak  {=  x)  to  be  generated  by 
a  point  moving  over  each  of  the  equal  distances  ah,  he,  cd,  etc., 


a       h        c        d  k 

Fig.  12. 

in  a  second  of  time.     If  ah  =  he  ^=  cd . . .  =■  h  in.,  the  rate  of 
change  of  x  is  h  in.  per  second.    Denoting  the  number  of  seconds 

*   |4,    called     factorial     4,    means     1x2x3x4.       In    general, 
|n=  1  X  2  X  3 .  .  .  X  «. 


46  DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

by  t,  we  have  x  =  hi.     Differentiating  this  equation,  regarding 
h  as  constant,  we  have 

dx 

clx  =  lidt,     and     -7—  =  h. 

Therefore,  I.  Iti  luiiform  cliange  the  increment  of  the  vari- 
able varies  as  the  increment  of  the  time. 

dx 
II.    The  derivative  -jj-  expresses  the  rate  of  cliange  of  x  ])er 

second,  ivhen  t  represents  seconds. 

TL.  Variable  Change.  When  a  variable  does  not  change 
nniformly,  its  rate  of  change  is  ever  changing,  and  is  measured 
at  any  instant  by  what  the  change  or  increment  would  be  during 
the  next  unit  of  time  if  it  (the  rate  of  change)  were  to  remain 
unchanged  during  that  time. 

Thus,  when  a  train  leaves  a  station  it  usually  goes,  for  some 
distance,  faster  and  faster;  that  is,  its  rate  of  change  or  velocity 
is  constantly  accelerated.  If  at  any  instant  during  this  time  we 
say  "  It  is  going  at  the  rate  of  20  miles  per  hour,^'  we  mean  that 
it  would  travel  that  distance  the  next  hour  if  its  present  rate 
should  remain  unchanged,  or,  as  in  uniform  change,  if  the  in- 
crement of  the  distance  should  vary  as  the  increment  of  the 
time,  that  is,  uniformly. 

The  same  is  true  of  any  other  variable;  that  is,  u  being  a 
function  of  time  {t),  the  I'ate  of  change  of  u  is  measured  by 
what  An  would  be  in  the  next  interval  of  time  if  it  varied  as 
At  or  dt;  but  du  is  what  Au  would  be  if  it  so  varied,  Art.  25, 
hence  the  differential  of  u  measures  its  rate  of  change. 

Cor.  I.  Since  du  <x  dt,  du  =  mdt, 

du 

dt   =  ■'"' 

where  w  (  =  -^  j  is  the  rate  of  change  of  u  per  second,  and,  be- 
ing a  function  of  t,  is  in  general  variable.  See  Appendix,  A^. 


SUCCESSIVE  BIFFEBENTIALS  AND  RATE  OF  CHANGE.   47 

EXAMPLES. 

1.  Eeqiiired  the  rate  of  change  of  the  area  of  a  square. 
Let  X  =  the  side  and  u  the  area;  then 

,  17         r.    7  du        ^  dx 

u  =  X  ,     and     du  =  2xdx.     .'.  -r--  =  Zx-rr- 

dt  dt 

[du  \ 
That  is,  the  rate  of  change  of  the  area  \-jt]  is  equal  to  the 

rate  of  change  of  the  side  {-jy  1  multiplied  by  twice  the  side  (2a;). 

We  may  omit  dt  and  regard  du  and  dx  as  the  measures  of 
the  rates  of  change  of  u  and  x.  Thus,  du  =  2xdx  signifies  that 
the  area  is  increasing  in  square  inches  2a;  times  as  fast  as  the 
side  is  increasing  in  linear  inches. 

2.  In  the  function  «  =  .i-"  —  4:X  +  5,  (1)  at  what  rate  is  u 
increasing  when  x  is  5  in.  and  increasing  at  the  rate  of  3  in.  per 
second  ?  (2)  At  what  rate  is  x  increasing  when  it  is  10  in.  and 
u  is  increasing  at  the  rate  of  40  in.  per  second  ?  (3)  What  is  the 
value  of  X  at  the  point  where  u  is  increasing  10  times  as  fast 
as  a;? 

Differentiating  the  given  function,  we  have 

du  =  {2x  -  4:)dx (1) 

In  this  equation  we  have  three  quantities,  viz. :  du,  the  rate 
of  increase  of  ?/;  dx,  the  rate  of  increase  of  x;  and  x;  hence, 
when  either  two  of  these  are  given  the  third  may  be  found. 

(1)  X  —  5  and  c/a;  =  3;  substituting  in  (1)  and  we  have 

du  =  (10  -  4)3  =  18,  Ans. 

(2)  du  =  40,  a;  =  10;  substituting  in  (1)  and  we  have 

40  =  (20  —  'i)dx,  whence  dx  =  2^,  Ans. 

(3)  du  =  lOdx;  substituting  in  (1),  we  have 

lOdx  =  {2x  —  4)f/a;,     or     10  =  2a;  -  4,     whence  x  =  T,  Ans. 

3.  If  X  increases  uniformly  at  the  rate  of  5  inches  per 
second,  at  what  rate  is  u  =  x^  —  4.a;  increasing  when  a;  =  10 
inches  ?  1480  in.  per  second. 


48  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

4.  In  the  function  y  =  2x^  +  6,  what  is  the  vahie  of  x  at 
the  point  where  y  increases  24  times  as  fast  as  a;  ?  x  =  ±  2. 

5.  If  the  side  of  a  square  increases  uniformly  at  the  rate  of  3 
inches  per  second,  what  is  the  length  of  the  side  at  the  time  the 
area  is  increasing  at  the  rate  of  20  sq.  inches  per  second  ? 

Let  X  =  the  side,  «-  =  the  area;  then  u  =  x^. 

6.  In  the  last  example,  supposing  the  area  to  increase  uni- 
formly at  the  rate  of  10  sq.  inches  per  second,  at  what  rate  will 
the  side  be  increasing  when  the  area  is  22  sq.  inches  ? 

Take  x  =  V^t-  ,—  in-  per  second. 

V22 

7.  A  circular  plate  of  metal  expands  by  heat  so  that  its  dia- 
meter increases  uniformly  at  the  rate  of  2  inches  per  second; 
at  what  rate  is  the  surface  increasing  when  the  diameter  is  5 
inches  ? 

Let  X  =  the  diameter,  u  =  the  area;  then  u  =  -x'^. 

bn  sq.  in.  per  second. 

8.  In  the  last  problem,  if  the  surface  increases  uniformly  at 
the  rate  of  50  sq.  inches  per  second,  at  what  rate  will  the  diam- 
eter be  increasing  when  it  becomes  5  inches  ? 

20  . 

—  m.  per  second. 

9.  The  volume  of  a  spherical  soap-bubble  iucreases  how 
many  times  as  fast  as  the  radius  ?  When  its  radius  is  4  in.,  and 
increasing  at  the  rate  of  ^  in.  per  second,  how  fast  is  the  volume 
increasing  ? 

Let  X  =  the  radius,  ti  =  volume;  then  m  =  ^tix^. 

(1)  4:7tx^  times  as  fast.     (2)  32;r  cu.  in.  per  second. 

10.  A  ladder  50  ft.  long  is  leaning  against  a  perpendicular 
wall,  the  foot  of  the  ladder  being  on  a  horizontal  plane  x  ft. 
from  the  base  of  the  wall.  Suppose  the  foot  of  the  ladder  to  be 
pulled   away  from   the  wall   at   the   rate  of  3  ft.  per  minute. 

(1)  How  fast  is  the  top  of  the  ladder  descending  when. r  =  14  ft.? 

(2)  How  fast  is  it  descending  when  a;  =  30  ft  ?     (3)  What  is  the 
value  of  X  when  the  top  of  the  ladder  is  descending  at  the  rate 


SUCCESSIVE  DIFFERENTIALS  AND  BATE  OF  CHANGE.   49 

of  4  ft.  per  miuute  ?  (4)  And  what  at  the  time  the  bottom  and 
top  of  the  hidder  are  moving  at  the  same  rate  ? 

Let  y  —  the  distance  from  the  base  of  the  wall  to  the  top  of 
the  ladder;  then  y  =  4/2500  —  x''- 

(1)  I  ft.  per  minute.  (•^)  2^  ft.  per  minute.  (3)  x  =  40  ft. 
(4)  X  =  25  ^2  ft. 

11.  AYhat  is  the  value  of  x  at  the  point  where  x^  —  5x^  -\-  17;c 
and  x^  —  3.1-  change  at  the  same  rate  ?  .r  =  2. 

12.  Find  the  values  of  x  at  the  points  where  the  rate  of, 
change  ©f  x^  —  12.t^  +  45x  —  13  is  zero.  x  =  o,  x  =  5. 

13.  In  a  parabola  whose  equation  is  y''  =  12.r,  if  x  increases 
uniformly  at  the  rate  of  2  in.  per  second,  at  what  rate  is  y  in- 
creasing when  X  =  3  inches  ?  2  in.  per  second. 

14.  In  the  same  parabola,  at  what  point  do  y  and  x  vary  at 
the  same  rate  ?  When  y  =  6. 

15.  In  the  ellipse  whose  equation  is  44|i/'^  -}-  25a;''  =  HHij 
at  what  point  of  the  curve  does  y  decrease  at  the  same  rate  that 
X  increases  ?  When  y  =  3  and  x  =  5^. 

16.  Find  the  points  where  the  rate  of  change  of  the  ordinate 
y  =  x^  —  Ga;'  +  3:^  +  5  is  equal  to  the  rate  of  change  of  the  slope 
of  the  curve.  Where  x  =  1  and  x  =  5. 

17.  Two  straight  roads  intersect  at  right  angles;  a  bicyclist 
travelling  the  one  at  the  rate  of  10  miles  per  hour  passes  the 
intersection  2^  hours  in  advance  of  another  travelling  the  other 
road  at  the  rate  of  8  miles  per  hour.  At  what  rate  were  they 
separating  (1)  at  the  end  of  1|  hours  after  the  first  man  passed 
the  intersection?  (2)  At  the  end  of  2^  hours?  (3)  Eequired 
the  distance  (y)  between  them  when  it  is  not  changing. 

(1)  ^  =  5^V;    C^)  ^  =  10;  (3)  y  =  Vt"-  Vll  mi. 

72.  Applications  to  Geometry.  The  rates  of  change  of 
the  areas  and  lengths  of  curves  and  of  the  areas  and  volumes  of 
surfaces  and  solids  of  revolution  are  given  by  Arts.  31,  32,  33, 
34.     The  applications  are  made  as  in  Arts.  62,  64,  65,  66. 


50  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 


r 


Rate  of  Change  of  Curves-     Formula,  ds  =  y  1-j-  ( y  j  dx. 

18.  Find   the   rate   of   change   of  the  arc  of  the  parabola 
—  Aax.  ds  =  y  l-\ dx. 

X 

19.  In  the  previous  example,  if  «  =  9,  and  x  increases  at  the 
rate  of  12  inches  per  second,  at  what  rate  will  the  arc  be  increas- 
ing when  X  =  16  inches  ?  ds  =  15  in.  per  second. 

20.  Show  that  the  rate  of  change  of  the  arc  of  the  circle 

^o  .     -y  Rdx 

a;^  -j-  w^  =  E"  IS  ds  —  — — 

21.  In  a  circle  whose  radius  is  20  in.,  the  abscissa  changes 
at  the  rate  of  n  in.  per  sec;  at  what  rate  is  the  arc  increasing 
when  x  =  12  in.  ?  ^n  in.  per  second. 

73.  Given  the  rate  of  change  of  the  are  of  a  curve  (ds), 
to  find  the  rates  of  change  of  its  co-ordinates  x  and  y. 

Take  the  parabola  y^  =  -iax,  then  ydy  =  2adx;  between  this 
equation  and  ds''  =  dx''  +  dy''  eliminate  (1)  dx  and  (2)  dy,  and 
we  have 

dy  =  — — -  ds      and     dx  =  y  — ^ — ds. 

\/4.ce-\-f  '    a  +  x 

22.  In  the  parabola  y"  =  4:X,  if  the  arc  increases  uniformly 
at  the  rate  of  5  inches  per  second,  at  what  rates  are  y  and  x  in- 
creasing when  x  =  9  inches  ?  

^  i/lO  and  I  |/10  inches  per  second. 

23.  If  the  arc  of  the  circle  x''  -{-  y'^  =  100  increases  at  the 
rate  of  5  inches  per  second,  at  what  rates  are  y  and  x  changing 
when  X  =  6  inches  ?  —  3  and  -f  4  in.  per  second. 

APPLICATION   TO   MECHANICS. 

74.  Velocity  is  the  rate  of  change  of  the  distance  described 
by  a  moving  body.  Hence,  if  s  =  the  distance,  v  =  the  velocity 
and  t  =  the  time,  we  have 

I.  V  =-71-;     .'.  s  =       vdt,    and     t  =   I  — . 

dt  *J  ^    V 


SUCCESSIVE  DIFFERENTIALS  AND  BATE  OF  CHANGE.   51 

Again,  denoting  the  rate  of  change  of  the  velocity  by  a',  we 
have 


II. 


,    civ    d-s  Cm     ^  ^    r^^^ 

a'  —  -^  =  -Tpr;     .-.  V  =  I  a' at,    and    t  —  I  -j . 
dt       dt  J  */   a 


EXAMPLES. 

1.  If  8  =  W,  what  is  the  velocity  and  its  rate  of  change  ? 

ds 

Since  s  =  W,  -^  =  Qt  =^  v,  the  velocity; 

dv 

and  since  v  =  6i,  ~J7  ~  ^    ~  ^''  ^^-"^  ^'^^®  '^^  change  of  ik 

Thns,  if  the  nnit  of  s  is  one  foot,  and  the  unit  of  t  one 
second,  v  =  -^  [=  6i]  ft.  per  second,  and  rt'  =  — -  [=  6]  ft.  per 

second. 

2.  A  body  passes  over  a  distance  of  ct^  in  f  seconds;  find  v 
and  a',  (1)  in  general,  and  (2)  at  the  end  of  9  seconds. 

(1)  -^_  and ^;  (2)  ~  and  -  -4- 

2Vt  4.Vf  6  108 

3.  A  body  after  moving  i  seconds  has  a  velocity  of  3^^  -|-  2t 
ft.  per  second;  find  its  distance  from  the  point  of  starting. 

s  ^fvdt  =  f  +  t\ 

4.  The  velocity  of  a  body  after  moving  t  seconds  is  5f  ft. 
per  second;  (1)  how  far  will  it  be  from  the  point  of  starting  in  3 
seconds  ?  (3)  In  what  time  will  it  pass  over  a  distance  of  360 
feet?  (1)  45  ft.;  (2)  6  sec. 

5.  A  body  moves  from  A,  and  in  t  seconds  its  velocity  is  14;!  ft. 
per  second;  (1)  how  far  is  the  body  from^  ?  (2)  In  how  many 
seconds  will  the  body  have  gone  847  feet  ? 

(1)  7f  ft.;  (2)  11  seconds. 

75.  The  velocity  is  positive  or  negative  according  as  6?  is  in- 
creasing or  decreasing,  and  a'  sustains  the  same  relation  to  v, 


52  DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

therefore,  if  s  increases  as  the  moving-  body  goes  forward  and 
decreases  as  it  goes  hacTcward,  the  body  is  moving  forward  or 
backward  according  as  v  \%  positive  or  negative. 

6.  A  train  left  a. station  and  in  t  hours  was  at  a  distance  of 
\f  —  M^  -(-  ]6^'  miles  from  the  starting-point;  required  the 
velocity  and  its  rate  of  change,  also  when  the  train  was  moving 
backward,  Avhen  the  velocity  or  rate  per  hour  was  decreasing, 
and  the  entire  distance  travelled  in  10  hours. 

s  ^^  \t^  —  4:t'^  +  IGt''  —  the  distance  from  station. 

d9 

Y  =  t'  -  12f  -{-32t  =  v  =  the  velocity. 

dv 

—  =  3f  -  2it  +  32   =  a'  =  the  rate  of  change  of  v. 

The  roots  of  P  -  ]2f  +  32t  ^  0  are  0,  4  and  8;  therefore  v 
is  negative,  and  the  train  was  moving  backward  from  the  4tii 
to  the  8th  hour. 

Again,  the  roots  of  3f  —  24:t  +  32  =  0  are  1.7  -  and  6.3  + ; 
hence  a/  is  negative,  and  therefore  v  was  decreasing,  from  the 
l.Ttli  to  the  6.3tli  hour. 

The  roots  of  it'  -  4f  +  16f  =  0  are  0,  0,  8  and  8;  that  is, 
s  =  0  when  t  =  S;  hence  the  train  was  at  the  starting-point  at 
the  end  of  8  hours,  having  gone  backward  as  far  as  it  had  forward. 

Since  the  train  moved  forward  the  first  four  hours,  then 
backward  the  next  four  hours,  and  then  forward,  the  entire  dis- 
tance passed  over  in  10  hours  was 

4  ^8  10 

r^'l  -  Is]  4-  [si   =  64  +  64  +  100  =  228  (miles). 

0  4  8 

7.  A  train  left  a  station  and  in  t  hours  was  moving  at  the 
rate  of  t^  —  21f  -|-  80/  miles  per  hour;  required  (1)  the  distance 
from  tbe  starting-point;  (2)  when  the  train  was  moving  back- 
ward; (3)  when  its  rate  per  hour  was  decreasing;  (4)  when  the 
train  repassed  the  station;  and  (5)  hcfw  far  it  had  travelled 
when  it  passed  the  starting-point  the  last  time. 


SUCCESSIVE  DIFFERENTIALS  AND  RATE  OF  CHANGE.   53 


(1)  s  =fvdt  =/{t"  -  21f  +  80^^/2^  =  \t'  -  7f  + 


40/^ 


(2)  From  the  5th  to  the  16th  hour;  (3)  from  the  2.37th  to 
the  11.72th  hour;  (4)  in  8  and  20  hours; 

(5)      [si-  \sf+  Uf=  4658+  miles. 

^  ■-       0  5  16 

8.  A  traveller  left  a  point  A  at  12  m.,  and  in  t  hours  after 
his  rate  per  hour  was  b  —  t  miles;  (1)  how  far  forward  did  he 
go  ?  (2)  At  what  times  was  he  8  miles  from  A  ?  (3)  What  were 
his  rates  per  hour  when  at  a  distance  of  10^  miles  from  A  ? 

(1)  12i  miles;  (2)  2  p.m.  and  8  p.m.;  (3)  +2  mi.  and  -  2  mi. 
per  hour. 

76.  Uniformly  Accelerated  Motion  is  that  in  which  the 

rate  of  change  of  the  velocity  («')  is  constant.    That  is,  v  changes 

/Iv       dv 
uniformly  or  v  a  /;  hence,  Art.  70,  — -  ^  -jr  =  a'  =  the  rate  of 

change  of  the  velocity. 

Formulas.         v  =J  a'dl  =  a't  -\-  C  —  a't  -\-v^,      .     .     (1) 

and  s  =fvdt  =J  {a't  +  v;)dt  =  ^a'f  +  vj  +  s„,    .     (2) 

in  which  v^  and  s^  represent  the  initial  velocity  and  distance; 
that  is,  the  values  of  v  and  s  when  ^  =  0. 

If  Vp  =  0  and  5„  =  0  when  ^  =  0,  then  (1)  and  (2)  become 

/2s  

V  =  a't,     s  =  ia't';     .\  t  =  y  —  and  v  =  V2a's.       .     (3) 

77.  The  increment  of  v,  or  acceleration,  produced  by  gravity 
is  about  32.17  ft.  per  second,  and  is  usually  represented  by  ^. 
Hence,  by  substituting  g  for  a'  in  (3),  we  obtain  the  four  for- 
mulas for  the  free  fall  of  bodies  in  vactio  near  the  earth's  sur- 
face. When  the  bodies  are  not  in  vacuo  the  formulas  generally 
are  slightly  inaccurate,  on  account  of  the  resistance  of  the  at- 
mosphere. 


54 


DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 


PROBLEMS. 

1.  A  rifle-ball  is  projected  from  0  in  the  direction  of  I'^vith 
a  velocity  of  b  ft.  per  second;  required  its  path,  knowing  that 

its  velocity  in  t  seconds  along  the  action-line  of 
gravit}^  {OX)  will  be  gt  ft.  per  second. 

Let   OX  and  OY  he  the  axis  of  x  and  y, 

respectively;  then  -~-  =  i,  and  -j-  =  gt; 

y  =  U,  .  .  .   (1)     and     x  =  ^jt\  .    .    (2) 
Eliminating  t  between  (1)  and  (2),  we  have 
,      25= 

y  =  — ^^ 

Fig.  13.  g 

that  is,  the  path  of  the  ball  is  an  arc  of  a  parabola. 

2.  A  body  starts  from  0,  and  in  t  seconds  its  velocity  in  the 
direction  of  OX  is  2aM,  and  in  the  direction  of  01^  is  a^f  —  ¥; 

find  its  velocity  along  its  path  Onm,  the 
distances  in  the  direction  of  each  axis  and 
along  the  line  of  its  path,  and  the  equation 
of  its  path,  the  axes  being  rectangular. 

Let  v-^,  Vy  and  t'g  represent  respectively 
the  velocities  in  the  directions  of  the  axes 
X  and  y  and  the  path  s.     Then 


(1) 


y  ^.ficef  -  lyu  =  \aH^  -  IHi  (2) 


'       cU        ^ 


=  VaH'  +  2a'dT  -\- b*  =  aH'  +  &^, 


s=J'{a'f+byit  =  iaH'-\-bH.    ...„.     =     o     (3) 

Now  to  find  the  path  of  the  body,  we  eliminate  t  between  (1) 
and  (2)  and  obtain 


CHAPTER  IV. 

GENERAL    DIFFERENTIATION. 

LOGARITHMS. 

78.  Lemma.  Tlie  limit  of  \l-\ — \  ,as  z  approaches i7ifinity, 

111 

is  the  sum  of  the  infinite  series  ^+7^  +  7^+77  +  ^^^« 

\:l      li      li 

Assuming  the  binomial  theorem  for  positive  integral  values 
of  z,  we  have 


which  evidently  approaches  ^  +  "kT  +  7^  +  ^tc,  as  z  approaches 

infinity.  The  sum  of  this  series  is  represented  by  e.  It  is  the 
base  of  the  natural  system  of  logarithms,  which  is  equal  to 
2.718281,  approximately. 

2  J-  CO  \  Z  J 


79.  To  differentiate  the  logarithm  of  a  variable. 

Let  u  =  loga  V,  where  ?^  is  a  function  of  x. 

55 


56  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

When  X  is  increased  by  h  we  have 

All  ~  \oga{v  +  ^v)  —  log„  V 

[v  +  Av  \ 


All 
A 

Passing  to  the  limit,  remembering  that  as  h  approaches  0, 
—  approaches  0,  and  -~r-  approaches  oo ,  we  have,  as  in  the  pre- 
ceding lemma, 

dn       1  ,  ,         /,        -.dv  * 

^  =  -  loga  e,    or    du  =  (]og„  e)--. 

KuLE. — Divide  the  differential  of  the  variable  by  the  variable 
itself,  and  multiijly  the  quotient  by  the  constant  loga  e. 

The  factor  logo,  e,  usually  represented  by  m,  is  called  the 

modulus  of  the  system  whose  base  is  a.     AVhen  a  =  e  the  mod- 

dv 
ulus  is  unity  and  we  have  du  =  — ,  simply.     Herein  lies  the 

advantage  of  the  natural  system  of  logarithms,  whose  base  is  e, 
in  all  discussions  of  a  theoretical  nature.  Hereafter  when  the 
base  of  a  given  logarithm  is  not  indicated,  it  is  to  be  understood 
that  the  base  is  e. 

EXAMPLES. 
1.  Differentiate  u  =  log(;?;'  —  2.^  +  5). 

_  d{x'  -  2x  +  5)  _  (3.y'  -  2)dx 

~     x'  -2x+5      ~  x'  -  2x  +  5" 

*  For  another  method,  see  Appendix,  A^ ,  Cor.  III. 


GENERAL  DIFFERENTIATION.  57 


2.  u  =  lo^f^^' 
^\1  —  X 


71  +  x\       1  +  X  2dx  1  —X         2dx 

3x"dx 


3.  u  =  log  i/'X^  —  a^'  du  — 


2{x'  -  a')' 


1       /-  „        3x«  7        4m(10  —  Sx)dx 

4.  ?^  =  logo  (oa;''  —  a; )  .  du  = n: ~—  . 

5x  —  X 

5.  u  =  log  (.r  +  Vl  +  a;^).  ^^ 


Vl  +  .T^ 


6.  w  =.  log  [(a  -  a;)  i/«  +  a:].  dii  =  -  -^-^K^^ 

7.  y  =  log\r.  dy  =  Slog'^a;— . 

4log^  (log.r)^/a; 
X  log  a; 

9.  y  =  log  (log*  x).  dy  =  — j^ . 


8.  y  =  log'  (log  x).  dy  = 


10.  y  ==  log  {x-\-a-{-  ^"lax  +  a;").  dy  = 

i'^2«.T  -{-  a;* 

EXPONENTIAL  FUNCTIONS. 

80.  To  differentiate  u  =  ^r". 
Passing  to  logarithms,  we  have 

log  11  =  V  log  «. 

Differentiating,  —  =  dv  log  a. 

Multiplying  by  tt,  du  =  u  log  a  dv. 

Substituting  a"  for  u,    du  =  a''  log  a  dv* 

*  For  another  metiiod,  see  Appendix,  Ai. 


58  DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

Hence,  tlie  differential  of  an  exponential  function  with  a 
confitant  base  is  equal  to  the  function  itself  into  the  natural 
logarithm  of  the  base  into  the  differential  of  the  exiionent. 

Cor.  I.  d{f)  —  e^dv,  since  log  e  =  1. 

EXAMPLES. 

1.  Differentiate  y  =  m^^^''^ . 

Ay  =  ,n  "^log  (m)a(  i/f+i^)  =  "' *''"^ '■"i^'^''^. 

2.  y  =  e'°^^.  -  dy  — . 

3.  ?/  =  e^  '°^  ^.  dy  =  e^  '°s  ^(log  x  +  l)dx. 

e^  e'^x  dx 

^■y  =  T+'x  '^y  =  -(i+xr- 

b.  y  =  e^(l  -  x").  dy  =  e^(l  -  dx'  -  x')dx. 


dx 


6.  ^/  =  log  — -- — -.  (^y  ~  1    ,     X- 

81.  To  diflerentiate  u  =  ^z*". 
Passing  to  logarithms,  we  have 

log  u  -—  V  log  y. 

^.n.         .    .  du       T    ,  ,     dii 

Differentiating,  —  =  dv  log  y  +  v-^. 

dii 
Multiplying  by  u,  du  =  u  log  ydv  +  uv^~. 

Substituting  y'"  for  «,      du  =  y''  log  ^/cZi'  +  mf'^dy. 

Hence,  the  differential  of  an  exponential  function  with  a 
variable  base  is  the  sum  of  the  results  obtained  by  first  differen- 
tiating as  though  the  base  were  constcmt,  and  then  as  though 
the  exponent  were  constant. 


GENERAL  DIFFERENTIATION.  59 

The  method  of  differentiating  a  function  by  first  passing  to 
logarithms,  as  in  the  two  preceding  demonstrations,  is  called 
logarithmic  differentiation.  It  may  be  used  to  great  advantage 
in  differentiating  many  exponential  functions  and  those  involv- 
ing products  and  quotients. 

EXAMPLES. 

1.  Differentiate  y  =  {x"  +  If+K 

dy  =  (.T^  +  l)-^i  log  {x^  +  l)d{x  +  1)  +  {X  +  l)(a;^  +  lYd{x'  +  1) 
=  {x-^  +  lYi{x'  +  1)  log  {x'  +  1)  +  2{x'  +  x)]dx. 

2.  y  =  a;^.  dy  =  x''{\og  x  -\-  l)dx. 

of/—-                            7         Vx  (I  —  log  x)dx 
6.  y  =  Vx  or  x^.  dy  = ^^ ^— ^ — - — . 

5.  ?/  =  e*  ,  dy  =  e"^  (1  +  log  x)x'^dx. 

Make  u  =  x"^,  differentiate,  and  replace  u  and  du  by  their 
values. 

6.  y  =  x''''.  dy  =  x^(---\-\ogx-\-  log'  x\xfdx. 

TRIGONOMETRIC  FUNCTIONS, 

82.    Circular   Measure   of  Angles,     li  v  =  the  length  of 
the  circular  arc  BP,  and  r  =■  the  length  of 
the  radius  OB  in  terms  of  the  same  unit,  the 

ratio  -  is  the  circular  measure  of  the  angle 

BOP.     When  r  =  1,  the  measure  of  the  an- 
gle is  simply  v,  which  is  the  length  of  the 
arc.     This  method  of  measuring  an  angle  is 
called  the  circular  or  analytic  system,  as  distinguished  from  the 
degree  or  gradual  method. 


60 


DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 


83.  To  differentiate  sin  v  and  cos  v. 

With  tlie  radius  OB  (=  1)  describe  the  circle  whose  centre 
is  0,  and  let  the  angle  BOP  or  its  measuring  arc  BP  be  any 
value  of  V,  which  is  a  function  of  x,  then 
PE  =  sin  V  and  OE  =  cos  v. 

Let  us  suppose  BP  or  v  to  receive  an  in- 
crement. Take  PT,  a  part  of  the  tangent 
line  at  P,  for  the  differential  {dv)  of  the  arc 
BP  (Art.  48),  then  the  proportional  incre- 
ments of  sin  V  and  cos  v  will  be  GT&nd  —  GP, 
respectively.  Therefore  GT  =  fZ(sin  v),  and 
-  GP  =  d{cos  v). 
The  angle  GTP  =  BOP  =  v;  hence  in  the  triangle  GTP 
we  have 

(1)  GT  =  cos  V  dv,         .:  d{sm  v)  =  cos  v  dv. 

(3)  GP  =  sin  V  dv,         .'.  f/(cos  v)  =  —  sin  v  fZw. 

Hence,  the  differential  of  the  sine  of  an  angle  is  the  cosine  of 
the  angle  into  the  differential  of  the  angle. 

The  differential  of  the  cosine  of  an  angle  is  minus  the  sine 
of  the  angle  into  the  differential  oj  the  angle. 

84.  Cor.  I.  TF  —  sin  {v  +  dv)  and  OF  =  cos  {v  -f  dv), 
which  approach,  respectively,  PTand  OB  as  v  diminishes,  and 
when  V  =  0  we  have  sin  dv  =  dv  and  cos  dv  =  1.  That  is,  the 
sine  of  the  differejitial  of  an  arc  is  the  differential  of  the  arc 
itself,  and  the  cosine  of  the  differential  of  an  arc  is  1. 

85.  The  differential  of  the  tangent  of  an  angle  is  equal  to 
the  square  of  the  secant  of  the  angle  into  the  diff'erential  of  the 
angle. 

sin  V 


For, 


tan  V  = 


d  tan  V 


cos  V 

cos  vd  sin  v  —  sin  vd  cos  v 
cos^*  V 

(cos''  V  +  sin^  v)dv<  „      , 

■^^ —^ —  =  sec  V  dv. 

cos  V 


(Art.  41) 


GENERAL  DIFFERENTIATION.  61 

86.  Tlie  differential  of  the  cotangent  of  an  angle  is  equal 
to  minus  the  square  of  the  cosecant  of  the  angle  into  the  diff'er- 
ential  of  the  angle. 

For,  cot  V  = . 

tan  V 

d  tan  V  sec^  vd  v  „      , 

d  coiv  = 7 — 5 —  =  — - — „ —  =  —  cosec"  v  dv. 

tan  V  tan  v 

87.  The  diff^erential  of  the  secant  of  an  angle  is  equal  to  the 
secant  of  the  angle  into  the  tangent  of  the  angh  into  the  differ- 
ential of  the  angle. 

For,  sec  v 


cos  V 


d  cos  V      sm  V  dv  .  , 

d  sec  V  = —  =  , —  =  sec  v  tan  v  dv. 

cos^  V  cos  V 

88.  The  diff^ercntial  of  the  cosecant  of  an  angle  is  equal  to 
minus  the  cosecant  of  the  angle  into  the  cotangent  of  the  angle 
into  the  diff'erential  of  the  angle. 

For,    cosec  v  = 


sm  V 


d  sm  V          cos  vdv  ,      , 

.*.     d  cosec  V  = --^ —  = r-T —  =  —  cosec  v  cot  v  dv. 

sm  V  sm   V 

89.  d  vers  v  =  f/(l  —  cos  v)  =  sin  v  dv. 

EXAMPLES. 

Differentiate  the  following: 

1.  y  =  sin  {x^  —  x). 

dy  =  cos  (a;^  —  x)d{x''  —  x)  =  cos  {x'  —  x){2x  —  1  )dx. 

2.  y  =  tan'  {x'). 

dy  =  4:  tan'  {x')d{tsin  x')  =  4  tan'  {x')  sec'  {x')d{x') 
=  12  tan'  {x')  sec'  {x')x'dx. 


62         DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

3.  y  =  cos  (ax).  dy  =  —  a  sin  {ax)dx. 

4:.  y  =  cos^  X.  dy  =  —  '5  cos'  x  sin  x  dx. 

5.  ?/  =  sin  x  cos  a;.  dy  =  (cos'  a;  —  sin'  x)dx. 

Q.  y  =  tan'  5^;.  dy  =  10  tan  (5a;)  sec^  {5x)dx. 

7.  y  =  sec^  .t.  dy  =  3  sec^  a;  tan  a;  dx. 

8.  ^  =  sin  a-  tan  a;.  dy  =  (sin  a;  +  tan  x  sec  a;)f/a;. 

9.  y  =  {x  cot  a:)'.  c??/  =  ^^  *^^^  a;(cot  a;  —  a;  cosec'  x)dx. 

10.  y  =  X  sin  a:  +  cos  a;.  dy  =  x  cos  a;  f/a;. 

11.  y  =  X  —  sin  a;  cos  a;,  <Z?/  =  2  sin''  x  dx. 

12.  y  =  tan  a;  —  x.  dy  =  tan'  a;  dx. 

13.  ?/  =  sin  (cos  x).  dy  =  —  sin  a;  cos  (cos  x)dx, 

14.  ?/  =  sin  X  —  I  sin'  a;.  dy  =  cos'  a;  c?a;. 

15.  y  =  ^  cos'a;  —  cos  x.  dy  =  sin'  a;  dx. 

16.  ?/  =  ^  tan^  a;— tan  a; -fa;,  dy  =  tan'  a:  dx. 

I      Sill     3y  "1 

17.  y  =  a;^i"^.  c?«/  =  a;^'°''    ■ — ^ \-  log  a;  cos  a;  \dx. 

18.  ?/  =  (sin  x)''.  dy  =  (sin  a;)''[log  sin  a;  +  a;  cot  a;]«fa-. 
Prove  the  following  by  differentiating  both  members  (see 

Art.  50) :  . 

19.  /  cot  X  dx  =  log  (sin  x)  +  C. 

20.  /  —  tan  a;  dx  =  log  (cos  x)  +  C. 

21.  /  sec  a;  cosec  a;  dx  —  log  (tan  a;)  +  C. 

22.  /  —  sec  X  cosec  a;  c?a;  =  log  (cot  x)  +  (7. 

23.  /  tan  x  dx  =  log  (sec  x)  -\-  C. 

24.  /  —  cot  xdx  =  log  (cosec  a-)  +  (7. 

/*  sin  a;  f/a;        ,       .  \    i    /^ 

25.  /  . =  log  (vers  x)  +  C. 

J   1  —  cos  X 

26.  /  sin  X  cos'  a;  c/a;  =  i  sin'  x  —  \  sin'  x -\-  C. 

27.  /  sin'  ;r  cos'  a;  c?a;  =  i  sin'  x  —  ^  sin'  a;  +  (7. 


GENERAL  DIFFERENTIATION.  63 


/sin   X  ^  ,  ,     _ 

— 7^ ax  =  sec  X  +  cos  x  +  6. 
COS"  X  ' 


28 

t/    COS" :?; 

Prove  the  following: 

29.  If /(a;)  =  {x-  -  6x  +  12)e^/"'(a;)  =  a;V. 

30.  If /(.'?;)  =  e~'^cosx,f^''{x)  =  —  46"^  cos  x 

31.  lif{x)  =  tsinx,f"'{x)  =  6  sec'  x  -Asec'x. 

„^    _„  7  cos  X        cos'  .'c    f?^?/        .   , 

32.  If^:=— ^ ^,-^=.sm^. 

33.  Iif{x)  =  x'\ogx,f^{l)  =  6. 

34.  If /(a;)  =  log  sin  x,f'"lj]  =  4. 

35.  lif{x)  =  sin  a:,  f'{0)  =  1;  /'''(O)  =  -  1. 

36.  It  fix)  =  log  (1  +  x),r'{0)  =  -  1;  /v(0)  :=  |4. 

37.  lifix)  =  «%/"(0)  =  log"  a. 

38.  If /(a;)  =  e^  log  .r,/^(.)  =  .flog  .  +  1  -  ^  +  ^  -  ^1. 

I 0         0  0  c 

INVERSE    TRIGONOMETRIC  FUNCTIONS. 
90.  To  differentiate  sin"^  y. 
Let  V  =  sin~^  y,     then     y  =  sin  v. 


^1/  =  cos  ?'  6?-y  =  Vl  —  ^Vy. 

,               di/                    7/  •    -1    \            ^^y*  • 
dv  =  —  ,     or     «(sin  ^  ?/)  =:  — —- 

91 .  To  differentiate  cos"^  y. 

Let  V  =  cos"-'  y,     then     y  =  cos  v. 


dy  =  —  sin  y  f/y  =  —  y  1  —  y^dv. 

dv  = ,    or     f/(cos""^  v)  = —- — , 

Vl  —  y''  Vl  —  y^ 

which  always  has  the  sign  of  —  sin  v. 

*  To  avoid  tlie  double  sign  ± ,  we  shall  suppose  0  <  «  <  — ;  for  any 
other  quadrant  the  sign  will  be  that  of  cos  v. 


64  DIFFERENTIAL  AND  INTEGRAL  CALCULU8. 

92.  To  differentiate  tan^^  y. 
Let  V  =  tan~^  y,    then     y  =  tan  v, 

cly  —  sec^  V  dv  =  (1  +  y'^)dv. 


T — f — r„     or     f/(tan~^  i/)  =  - — ;~ 


In  a  similar  manner  we  find 

93.                  ,Z(cot-^)  =  --^. 

Q4                        d(-oo-^  iA  —            -^ 

»-*.                      «^,bLO     y)  —            L^. 

yVy'-l 

95.                diGosec-^  y)  — — -. 

96                         /7^vrr"-i  ?y^  —              '^ 

»o.                 «^vcis    y)  —       ^_. 

EXAMPLES. 

Differentiate  the  following  : 

1. 

y  =  sin  ~^  Vx. 

dx 

__          dVx          _     3  4/^    _ 

dx 

^        ^l-(4/.^y         i/1-^ 

3  !/:»  -  a:'' 

2. 

y-^^''~ii+^- 

(l-  x\                         2dx 

^,,_    Vi  +  J     _           (i  +  xY 

^a:; 

1  +  x"' 

3 

-1                                                     7                    ^^^^ 

xVn'x'- 

-  1 

1 

,  2x                                       ,     •           dx 

^      "^         9-                                     ^^        ./a,_ 

r?;^ 

4/1  —  x' 
clx 


GENERAL  DTFFEBENTIATION.  Q5 

b.  y  —  sin~^  (3a;  —  4.r^).  dy  = 

Q.  y  =  sin-i  {2x  -  1).  dy  =      . 

Vx  —  X 

7.  ?/  =  sin"^  (sin  x).  dy  =  dx. 

8.  y  =  sin~^  (l^sin  .r).  dy  =  ^{Vl  -\-  coseGx)dx. 

o  ,     _,     2.T  ,  2dx 

9.  y  =  tan      3.  dy  = 


1  -  a;'  ^        1  +  a;' 


10.  y  =  tan-i  f|/L^-^^^')  rf^  =  ^dx. 

•^  V  ^    1  +  cos  x/  ^       ^ 


11.  y  =  (x  -\-l)  tan"'  Vx  —  i/a;.  fZy  =  tan"^  \^dx. 

Prove  the  following  by  differentiating  both  sides: 


^    a  -V  X        a  a 


.^      r       dx  1  X 

14.  / —  =  -sec  '  — V-  C. 

"■'  xVx^  -  a?      «  « 

15.  /*-^^  =  sin-i-+a 


a;" 


a 


16.  /"    ^      ^^ r=vPrs-i^+<7. 

17.  /  t^^^^^^^a;  =  ^_^[^^  _|_  ^  sin-i  -  +  C. 

18.  f-^^-=^  =  -  ^±^  4/2aa;  -  x'  +  |a^  vers-^  -  +  C. 

^    V'2«a;  -  a;^  3  a    ' 

19.  /  =  =  m  sin  M ■ —    +  C. 

^'   V&-lf-  2ahx  -  a^x""  V      c      ) 

20.  f  ''^^^^— =  m  log(«a:+&+4/c^+(^+aa;)'+  C 

r             macdx                                (ax  +  h\       ^ 
21-    /    ,  o   ,   ^  , — ,    ,„   . — r  —  m  tan  M ■ —     +  (^• 


66  DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

r    m{ad  —  hc)dx  (ax-^h\ 

J    {ax -^  l){cx  ^  d)  ^\cx-\-dJ    ' 

The  last  four  equations  may  be  conveniently  employed  in 
integrating  a  certain  important  class  of  differentials,  of  which 
the  following  are  illustrations : 

hdx 
33.   Eequired  the  integral  of  - 


-  3a;  -  28  • 

Here  x"^  —  ox  —  2Q  =  {x  —  7)  (a;  +  4) ;  hence  we  may  inte- 
grate by  formula  22,  thus :  Make  ax  -\-h  =  x—1,  ex  -\-  d  =  x  -{-  4, 
and  'm{ad  —  dc)  =  5;  we  then  have  a  =  1,  h  —  —  7,  c  =  l,  (Z  =  4 
and  m  =  y\;  hence,  substituting  in  22,  we  have 

/5dx  5  ,      /a;  —  7\    ,    ™ 

^■--3a;-28=ri^^gU-^'  +  ^- 

24.  Eequired  the  integral  of 


dx  -  28       11  ^'^'=  \x  +  4:J    ' 
3dx 


4:x'  +  3a;  +  1 
Integrating  by  Ex.  21,  we  have  a^  =  4,  2al)  =  3,  J'  +  c"  =  1 

and  mac  =  3  ;  whence  a  =  2,  i  =  h  c  =  i  V'7  and  m  =  —-. 

r         Mx         _    6  ,  {2x-\-l\ 

•  V  4a;^  +  3a; +1  -  :^  *^''     VTW ' '^ 

25.  Find  the  integral  of 


|/16  -  12a;  -  4a;' 


Integrating  by  Ex.  19,  we  have  a""  =  4,  2ah  =  12,  c^  —  W  =.  16 
and  ma  =  5;  hence  a  =  2,  b  =  3,  c  =  5,  m  =  |. 

r  Mx  5   .    _J2a;+3\   ,    ^ 

•••    /     ,      — :  -  —  77  sm    M — - —    +  ^• 

*^    l/l6  —  12a;  —  4a;'        3  \      o       / 

"  ■  t/  a;'  +  a;  +  1'  1/3  i/3 

„^      /*      ^?a;  1     ,       ,  a;  4/5.    ,    ^ 

27.    /  ^   ,   ^  ,.  — -  tan-^  — z^  +  G, 

J  2  + 5a;'  4/10  4/2 


GENERAL  DIFFERENTIATION. 


67 


r dx 

'  J  2x^ 

■f 

^         i/r. 


4:X 

dx 


V2  ,      2x  —  2  -  d  V2    ^    ^ 
£3-  log  :: .  ^  .,^-  +  G. 


-  2  +  3  ^2 


4/1  —  "bx  —  x^ 
dx 


sm' 


-1 1±|^  +  G. 


Vm  +  nx  +  ?■: 


-^  log  [x  Vr  +  ^]:  +  ^''i  +  nx  +  r^-'^j  +  a 


dx 


I    Zrx 


n 


Vm  +  nx 

dx 
m  -\-  nx-{-  rx^ 


rx' 


y  ;•  \  |/4mr  +  n' 

9 


+  a 


V4:mr  — 


n' 


tan-.(-|f-±iL,)+a 
r  4mr  — 


97.  To  find  the  differential  of  an  are  in  polar  co-ordinates. 

Let  AF  {=  s)  he  the  arc  of  a  curve,  0  the  pole,  OP  (=  r) 
the  radius  vector,  and  FT  a.  tangent 
to  the  curve  at  P. 

Let  6  =  XOF  and  tp=  OFT. 
Increase  6  by  FOF'  {=  /iO),  then 
arc  FF'  =  As  and  OF'  =  r  +  Z/r. 
Draw  FD  perpendicular  to  OF', 
then  FD  =  r  sin  J  6'  and  i>P'  = 
r  -j-  z/r  —  r  cos  z/^. 

The  chord  PP'  =  VP^M^i^P^, 

chord  PP 


J^ 


'      i/iPDV.lDF'V 


Passing  to  the  limit,  remembering  that  as  z^^  approaches  0, 

the  limits  of  Ti^F^r-  and  — -r-.-r-  (Art.  33),  also  of  cos  Ad,  is 

arc  FF'  A6     ^  "  ' 

each  unity,  we  have 

ds  — 


r^  + 


dd) 


68 


DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 


98.  CoK.  I.  As  P' approaches  P,  the  angle  OP'P  approaches 
the  angle  OPT  {=ip)',  therefore 


limit  rPZ>" 


rdd 
dr 


sin  ^  = 


rdd 
ds  ' 


dr 
cos  w  =  ^— . 
'^       ds 


FUNCTIONS  OF  TWO  OR  MORE  INDEPENDENT  VARIABLES. 


dy 


y 


99.  A  function  of  two  variables,  as  w  =  xy  +  y^  and 
?^  =  sin  {x  +  ?/)>  is  represented  hj  f{x,  y);  and  similarly /(a;,?/,2;) 
represents  a  function  of  the  three  variables  x,  y  and  z. 

Since  x  and  y  are  independent  of  each  other,  the  function 
u  =f{x,  y)  may  change  in  three  ways.    Thus,  let  u  =^xy  =  area 

of  OB  PA,  where  OB  =  x  and  OA  = 

y.     (1)    a;  may   change  and  y  not, 

which  would   give   du  =  BCDP  = 

i?yaa:;  (2)  ?/  may  change  and  x  not, 

which   would  give   du  =  APFG  = 

a;f/y;  (3)  both  a;  and  y  may  change, 

which  would  give  (understanding  by 

du  the  portion  of    the    increment 

which  is  of  the  first  degree  in  dx  and  dy)  du  =  ydx  -\-  xdy. 

Hence  du  may  have  three  different  values,  and  it  is  desirable  to 

employ  a  notation  by  which  they  may  be  represented. 

100.  A  Partial  Differential  of  a  function  of  two  or  more 

variables  is  the  differential  obtained  on  the  hypothesis  that  only 
one  of  its  variables  changes,  as  ydx  and  xdy  in  the  previous 

du  ,        ,  dii  -, 

Ay, 


dx 


X 

FiCx.  18. 


example,  and  these  are  denoted  respectively  by  -j-  dx  and 
or  dyU  and  d,,u. 


dy 


101.  Total  Differential.  In  a  function  of  two  or  more  in- 
dependent variables,  if  each  variable  receives  an  increment,  that 
portion  of  the  corresponding  increment  of  the  function  which  is 
of  the  first  degree  with  respect  to  the  in-crements  of  the  variables 
is  the  total  differential  of  the  function. 


GENERAL  DIFFERENTIATION.  69 

102.  Peop.  The  total  differential  of  a  function  of  two  or 
more  independent  variables  is  the  sum  of  its  partial  differentials. 
Let  u  represent  any  function  of  x  and  y. 
When  X  becomes  x  -\-  h,  the  part  of  the  corresponding  mcie- 

ment  of  u  which  involves  the  first  power  of  h  or  dx  is  -j—dx. 

Hence,  omitting  the  terms  involving  the  higher  powers  of  dx, 
Art.  39,  the  new  value  of  u  is 

u  -\-  -i—  dx. 
dx 

In  this  new  value  of  u  when  y  is  increased  by  ^  (=  dy')  the 

parts  of  the  corresponding  increments  of  u  and  -j-dx  which  in- 

volve  only  the  first  power  of  dy  are  -j-dy  and  -r-(-— ffe  ]^?/; 

hence,  omitting  the  terms  involving  the  higher  powers  of  dy, 
the  second  new  value  of  u  is 

,  dii  -.     ,   da  ,     ,     d  Idu  .  \  ^ 

which  result  is  the  same  as  if  x  and  y  had  changed  simulta- 
neously, for  the  result  of  increasing  x  by  dx  and  y  by  dy  is  evi- 
dently the  same  whether  the  changes  be  made  separately  or 
simultaneously.  Hence,  since  the  last  term  involves  the  product 
of  dx  and  dy,  we  have 

,         du  ,     ,   dio  -, 
au  =  -=— «a;  +  -j-dV' 

In  a  similar  manner  it  may  be  shown  that  the  theorem  is 
true  of  functions  having  any  number  of  variables. 

CoK.  I.  The  total  differential  of  a  function  is  the  sum  of 
those  parts  of  its  increment  which  vary  as  the  increments  of  the 
variables,  respectively. 

CoK.  II.  The  theorem  is  also  true  of  functions  whose  variables 
are  not  independent  of  each  other. 


70  DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

CoE.  III.  The  total  differential  of  a  function  of  two  or  more 
variables,  as  x,  y  and  %,  is  the  sura  of  the  differentials  obtained 
by  first  differentiating  as  though  x  only  were  variable,  and  then 
as  though  y  only  were  variable,  and  then  as  though  z  only  were 
variable. 

103.  A  Partial  Derivative  of  a  function  of  two  or  more 
variables  is  the  ratio  of  the  partial  differential  of  the  function  to 
the  differential  of  the  variable  supposed  to  change. 

104.  The  Total  Derivative  of  a  function  of  two  or  more 
variables,  only  one  of  which  is  independent,  is  the  ratio  of  the 
total  differential  of  the  function  to  the  differential  of  the  inde- 
pendent variable. 

To  prevent  confusion  we  shall  distinguish  the  total  differen- 
tial or  derivative  of  two  or  more  variables  by  enclosing  it  in 
brackets. 

Thus,  if  u  =f{x,  y),       {chi]  =  —dx+  j-dy, 

du  du 

where  -j—dx  and  -r-f??/  are  the  partial  differentials  with  respect 

to  X  and  y,  respectively. 

EXAMPLES. 
Find  the  total  differential  of — 
1.  u=x^  —  Zxy  +  ^f. 

■—dx  =  {2x  —  ^y)dx;     -T-dy  =  —  (3a;  —  4:y)dy. 
cix  cty 

.-.  [dn]  =  {2x  —  ?,y)dx  —  {^x  —  4:y)dy. 

X  -y  {x  -y) 

.     ,x  ,  ,  ,       ydx  —  xdy 

y  y  ^y""  —  ^' 

4.  u  =  tan-^  ^.  [dit]  =  ^^  {  ~  ^. 

X  :  a   +  y 


GENEBAL  DIFFERENTIATION.  71 

5.  n  =  log  y^.  [^cIil]  =—dy-{-  log  ydx. 

6.  u  =  y'^  ^.  [chi]  =  y''''  ^  log  y  cos  x  clx  +  -^^  c?^/. 

105.  Function  of  Functions.  If  it  =  F{y)  and  ?/  =f(x),  u 
is  indirectly  a  function  of  x  through  y.     In  such  cases  the  value 

(lit 

of  y—  may  be  obtained  by  finding  the  value  of  u  in  terms  of  x, 

and  differentiating  the  result;  but  it  is  often  more  easily  found 
by  the  formula 

du  _  du  dy 

dx  ~  dy  dx ' 

That  is,  the  derivative  of  u  with  respect  to  x  is  equal  to  the 
derivative  of  u  with  respect  to  y  multiplied  by  the  derivative  of 
y  with  respect  to  x. 

Thus,  if  u  =  tan~^  y,  and  ?/  =  log  x,  then 

du  _       1  dy  _1  J      .  '^^  _         1 

dy  ~  l-^-y"^'    dx  ~  x'  '  '  dx       x{l  -j-  y"") ' 


If  ic  =  F{v,  y),  V  =  f{x)  and  y  =  f^[x),  to  obtain 


du 
dx 


,  the 


total  derivative  of  u  with  respect  to  x,  we  may  proceed  thus: 

Since  \dti\  =  -^dv-\-  -j-dy, 

^     ^       dv  dy  -^ 


dividing  by  dx, 


'du 
dx 


du   dv       du  dy 
dv  dx       dy  dx' 


/Y/l/ 

which  gives  y-  in  terms  of  derivatives  which  can  be  reckoned 

out  from  the  given  equations. 

Thus,  if  u  =  v^  -\-  vy,  v  —  log  x  and  y  =  &%  then 

du      ^     ,  du  dv       1  ,  dy 

-J-  =  2v  -\-y,    —-  =  V,    -J-  =—     and  -~  =  e^; 

dv  ^      dy  dx       x  dx 


72  DIFFERENTIAL  AFD  INTEGRAL  CALCULUS. 

substituting  in  (2),  we  have 


du  _  2v  -\-  y 


-\-  ve^ 


dx  X 

If  ti  =  F{x,  V,  z),  V  =/{x)  and  z  =fX^),  we  have 

[dn] 


du  ,     ,     d^^,  ^     ,   du  , 
-^—  dx  -\ — —dv  +  -^dz. 
dx  dv  dz 


'du 
dx 


du        du  dv      du  dz 
dx        dv  dx      dz  dx ' 


(3) 


,        du    du       ^  du  J.-  1    T     •     ..  T    rdu^  ,, 

wnere  -y-,  -j-  and  -—  are  partial   derivatives,  and      ^—     the 
dx    dv  dz  ^  \_dx  J 


du    du       ,  du 
a 
total  derivative  of  u 


EXAMPLES. 


Find 


y-     in  the  following: 


1.  21  =  e^'d/  —  z),  y  =  sin  x,  and  z  =  cos  a;. 

'  dti 
dx 

'du' 

.  and  ?/  =  y?-"  —  a;\ 

X' 


2.  u  =  tan~^  ",  and  y  =  Vr' 


3.  M  =  tan  ^  (a;?/),  and  y  =  e^. 


dx 

'du' 

dx 


=  Seisin  a;. 
1 


Vr'  -  x' 
e^(l  +  x) 


4.  If  _-?/  =  uz  and  ?;  =  e^,  ^  =  a;*  —  4a;^  +  12a;^  —  24a;  +  24, 
find  the  slope  of  the  curve  of  which  y  is  the  ordinate  and  x  the 
abscissa.  dy 

dx 


e^x 


106.  Successive   Partial    Differentials   and   Derivatives. 

We  have  seen  that  the  differential  of  2i  {  =  f{x,  y) )  (1)  with  re- 
spect to  X  is  denoted  bj  -j-{u)dx,  and  (2)  with  respect  to  y  by 


dx 


-^{-)dy. 


GENERAL  DIFEERENTIATION.  73 

Similarly,  the  difPerential  of  -j^dx 

(1)  with  respect  to  x  is  denoted  by  -j-(—j—dx  jdx,  =  — j-^ — ; 

(2)  with  respect  to  y  is  denoted  by  ~{~j~dx\dy,  =  —  ■. 

(ty  \  (tx       I  iix  ciy 

d^  H  dii  dx 
Hence  — p  -^^ —  is  a  symbol  for  the  result  obtained  by  dif- 

dy  dx  •'  -^ 

ferentiating  n  two  times  in  succession :   once,  and   first,  with 

respect   to  y,   and   then   once   with    respect   to   x.     Similarly, 

d^u 
-^ — ^--idx  dy'^  indicates  the  result  of  three  successive  differentials 

of  u\  first,  once  with  respect  to  x,  and  then  twice  with  respect 
to  If. 

In  finding  these  successive  partial  differentials,  we  treat  dy 
and  dx  as  constants,  since  y  and  x  are  regarded  as  independent 
variables,  see  Art.  68. 

The  symbols  for  the  partial  derivatives  are 

d\i       d'^u        d^u      d^u  d^u 

d^'    cWdy'     df'    dx^'    d^~d^' 

107.  Principle.  If  u  =  fix,  ii),  - — 7-  =  -, — r— 

^     dxdy      dydx 

d^u 
For,  Art.  102,  changing  x  and  then  y  to  obtain  - — —,  or 

d~  1.1 
changing  y  and  then  x  to  obtain  - — 5-,  is  equivalent  to  changing 

dy  clx 

X  and  y  simultaneously;  and  therefore  the  results  are  equal. 

Cor.  I.  If  u  be  differentiated  m  times  with  respect  to  x,  and 

n  times  with  respect  to  y,  the  result  is  the  same  whatever  be 

the  order  of  the  differentiations. 

EXAMPLES. 

1.  Given  u  =  x^ii^ ;  find  -^ — ^  and  ^ — ^.  6x?/*. 

^  dxdy         dydx  "^ 


74  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

2.  Griven  u  =■  x'y  +  xy'\  verify  - — y-^  =   ,  „  -,  . 

d^ii         d^u 

3.  li  u  =  y  log  (1  +  xy),  show  that 


dy  dx      dx  dy 


.     ri-  xy—  If  ..      d^u  d'^u 

4.  (riven  w  —    .,       \  ;  verify 


^■"  +  y^ '  ^^  ^y     <^y  ^^ ' 

108.  To  find  the  successive  dilferentials  of  a  function  of 
two  independent  variables. 

Let  u  =  /{x,  y) ;  then 

[_du]=^JLdx+^dy (1) 

Differentiating  (1)  and  observing  that  -^  and  -7-  are  usually 

functions  of  x  and  y,  and  that  x  and  y  are  independent,  Art.  68, 
we  get 

[d^u]  =  ^dx^  +  ^dx  dy  +  ^/y  dx  +  ^df, 
or,  Art.  107, 

L<fu]=^<^.'+2^^S.ay+plf (2) 

Differentiating  (2),  remembering  that  each  term  is  a  function 
of  X  and  y,  we  have 

Id^u]  =  -y^dx"  4-  3  .,  ^  ^  dx""  dy  +  3^ — ^„dx  dv^  +  -r^dii^  .  .  ,  ; 
^       ^       dx^  '      dx^  dy         •'    '     dxdij         ^        «^ 

and  similarly  may  [c?^;],  [(?^?<],  etc.,  be  found.  By  observing 
the  analogy  between  the  values  of  \(P\l\  and  [f^w],  and  the  de- 
velopments of  {a  4-  a;)^  and  {a  +  xf,  the  formula  for  the  value 
of  [(/"?;]  may  be  easily  written  out. 

109.  Implicit   Functions.    In    functions     of     the     form 

du  dxi 

f{x,  y)  =  0,  the  formula  [du]  =  ;r^dx  +  3~^2/  is  often  useful  in 

ax  cii/ 

'        dv 
finding  the  value  of  the  derivative,  or  slope,  -,-. 


GENERAL  DIFFERENTIATION.  75 

Thus,  take        f{x,  y)  —  ax^  +  a;  sin  ?/  =  0. 
Making  u  =  ax^  +  x  sin  y,  we  liave 

\du'\  =  -n—dx  +  -^dy  =  (^ax^  +  sin  ?/)^a:  +  x  cos  2/  c??/. 


But  since  u  =  0,  [clu]  =  0. 

du 

dy  _        dx    __       3ax^  +  sin  y 

dx  ~       du    ~           X  cos  y 

dy 

EXAMPLES. 

dv 
Find  the  derivative,  -^,  of  the  following: 

1.  {y-hr-x^  +  ax^  =  0.                             ^  = 

3a;'  —  2a;a; 
2{y  -  b) ' 

2.  a;^  +  2ax'y  -  ay'  =  0.                                p- = 

Ax'  -\-  4axy 
day'  -  2ax^ ' 

S.  x'  +  ^axy+f  =  0,                             ,       ^  = 

x'  +  ay 
y'  +  ax 

110.  Successive  Derivatives    of    an  Implicit  Function. 

The  following  examples  will  serve  to  illustrate  how  the  succes- 
sive derivatives  of  implicit  functions  in  general  may  be  deter- 
mined. 

EXAMPLES. 

1.   Find  j^3  when  y''  —  Aax  =  0. 

XT  ^y      2a 

Here  -f^  =  — (1) 

ax         y  ^  ' 

Differentiating  (1),  we  have 

d^y   _  —  ^ady  . 


76  DIFFERENTIAL  AND  INTEGRAL   CALCULUS, 

Eliminating  dy  in  (1)  and  (2),  we  have 

dx'  y* ' 

2.  Given  a""?/^  +  Ifx"  —  nW  =  0,  show  that  3:^  = ^-3- 

-^     '  dx  ay 

3.  Given  tf  +  x'-  ^axy  =  0,  show  that  ^  =  -  -^^-^3. 

111.  Change  of  the   Independent  Variable.     After   ob- 
taining the  derivatives  -—,  —,  ^j\,  etc.,  on  the  hypothesis  that 

X  was  the  independent  variable  and  y  the  function,  it  is  some- 
times desirable  to  change  these  expressions  into  their  equivalents 
with  y  for  the  independent  variable  and  x  the  function,  or  with 
x  and  y  for  the  functions  and  some  other  variable,  as  t,  for  the 
independent  variable,  and  so  on. 

112.  To   find   the   successive   derivatives    of    -j-  wnen 
neither  x  nor  y  is  independent. 

Under  this  hypothesis  ^  is  to  be  differentiated  as  a  fraction 

having  both  terms  variable. 

d^ll  _  d  fdy\  __  dx d^y  —  dyd^  ,y. 

~dx^  ~~  dxXdxJ  ~  dx'  


Similarly,  d?  =  dx\d^') 

_  (dx  d'l/  —  dy  d^x)dx  —  ?,{dx  d^y  —  dy  d''x)d''x  ,^. 

~  dx" 

In  like  manner  we  obtain  the  other  successive  derivatives. 
OoK.  I.  If  y  is  independent,  then  d'^y  —  d^y  =  0,  and  we  have 


d^y  dy  d'^x 

dx^  "         dx^  ' 


(3) 


d\i      Zid^xfdy  —  d^x  dy  dx  ... 

^=  1^^  ^^^ 


GENERAL  DIFFERENTIATION.  77 

Formulas  (1)  and  (2)  give  us  the  values  to  be  substituted  for 

-j^  and  -v"!  when  neither  x  nor  y  is  independent;  and  formulas 

(3)  and  (4)  give  us  the  values  of  the  same  derivatives  when  y  is 
independent. 

If  a  new  variable  t,  ot  which  x  =  f{t),  is  to  be  the  inde- 
pendent variable,  in  Art.  Ill,  we  replace  x,  dx,  d'x,  etc.,  by 
their  values  as  determined  from  x  =  f{t). 

EXAMPLES. 

1.  Given  y(Vy  -\-  dy""  +  f^^''  =  0,  where  x  is  independent,  to 
find  (1)  the  transformed  equation  in  which  neither  x  nor  y  is 
independent;  also  (2)  the  one  in  which  y  is  independent. 

(1)  Dividing  by  dx^,  substituting  for  -—_  from  (1),  and  multi- 
plying both  members  by  dx^,  we  have 

y{d^y  dx  —  d^'x  dy)  -\-  dy"^  dx  -\-  dx^.—  0. 
(2)  Making  d"y  —  0  in  this  last  equation,  and   dividing  by 
—  dy^,  we  have 

d^x      dx^      dx  _ 
-^ dy""      dy^      dy  ~ 

2.  Change     the     independent    variable    from    x    to    t    in 

-74+-T^  +  y  =  0,  when  x  =  2Vt. 
dx       xdx 

Substituting  for  -~  from  (1),  multiplying  by  xdx^,  and  mak- 
ing X  =  2Vt,  dx  =  t'^dt,  and  d^x  =  —  ^f^df,  we  obtain 
^  d'^il      dy   , 

Change  the  independent  variable  from  x  to  y  in  th€  two  fol- 
lowing equations : 

Wa;7        dxdx^      dxAdx)  '     '     dy^      dy"" 

d^y      dy^      dy  _  d^      dx"       ^  _  ^ 

dx''      dx^  ~  dx  ~    '  dy''      dy^  ~ 


78  DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

Change  the  independent  variable  from  x  io  t  in.  the  two  fol- 
lowing equations: 

^    cf'y  ^      2x     dy  ^         y  ^      ^  ,       . 

5.  — ^  +  — - — I ~-  +  ,-,    ,     .,.2  =  0,  where  x  =  tan  t. 
dx"  ^   1  +  x"  c?a;  '  (1  +  x^Y 

^%i        dv  d'^v 

^'  (^  -  ^^te  -  ''dx  =  ^'  ""^''^  ^  =  '''  ^-  J  =  ^- 

7.  Find  the  value  ot  E  =  f  1  +  -—j   -^  —., ,  where  x  is  in- 
dependent, supposing  neither  x  nor  ?/  to  be  independent. 

_  _{da^dff_ 
dx  d'^y  —  ^j/  t^'^a;' 

MISCELLANEOUS  EXAMPLES. 
Differentiate  the  following: 


1.  y  =  3(a;^  +  l)'(4'c^  -  3).  dy  =  56x\x'  +  l)'^a;. 

1  dy  X 

2.  y  = r.  y^  =    ^  -  1. 


^       _Va  -\-x  -{-  ^a  —  X  (^y  _      a^-\-aVa'^—x^ 

""'  y  ~  |/^qr^  _  ^^zr^  dx  ~      xWa'-x" 


4.   y  =  X  -{-  log  COS  f-  —  x\ 


dy  2 


dx       1  -|-  tan  X 

_        Va  -\-  Vx  ^  _         l^« 

a;  log  a;   ,  T       ,,          .  dy         log  a; 

i.  y  -  log  r  ^,-^  ^  ^  ^.  ^^^  -  2,4  _^  a;^  _^  r 

8.  2/  =  sin  (a;  +  «)  cos  {x  —  a).  dy  =  cos  2xdx. 

9.  y  =  logtan(^+-).  ^    ^=^^«^-- 

10.  sin  2a;  =  2  sin  a;  cos  a;.  cos  2a;  =  cos"  x  —  sin''  x. 


GENERAL  DIFFERENTIATION.  79 

- ,      .     _  2  tan  X  ^        1  —  tan°  x 

11.  sm  2x  =  - — -— — ^— .  cos  2x  =  - — -— — -^— . 

1  +  tan  X  1  +  tan  x 

13.  sin  3z  =  3  sin  a;  —  4  sin^  x.       cos  3.'^  =  4  cos^  x—o  cos  x. 


1  a» 


dy  1 


13.  y  =  tan"^  e .  ,    _  . 

Jx' -  1\  dy  Qx" 

14.  ?/  =  cos"M-;r-; — -  .  -^  = ■„ . 

•^  \x'  +  i;  ^a;  x'  +  1 

^  „  ,1  dy  2 

lo.  y  =  sec"^ -— 5 r.  -—= == 

/^  3a;'  —  1  dx  \/i  _  . 

16.  y  =  tan"^  a;  +  tan~^  - — , — .  -^  =  0. 

1  -j-  X  dx 


Prove  the  following  by  differentiation: 

17.    /  — 5 ; — -5—^-^; — =  tan~^  (n  tan  x)  4-  C. 

J   cos  X  -\-  n   sm  o:  ^  '    ^ 

r2ax^dx  _^x       I         (x  —  a\    ^    ^ 

19.  log  (1  +  .r)  =  .r  -  ^4-'^-^  +  etc. 

X^       X^        x' 

30.  tan"^  X  =  x  —  —-{-—  _'—  -j-  etc. 
6        5  i 


Find  the  slopes  of  the  following  curves: 
21.  The  quadratrix,  ^  =  [a  —  x)  tan  ^r— . 


dy  TT     .  ^  „   TtX         ^  TtX 

dx        2a  ^  '         2a  2a 


23.  The  cycloid,  x  =  r  vers"^  -  —  \/2ry  —  y". 


dx  y    ' 


so  DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

c  i  -        --\  dii       1/  -        --\ 

23.  The  catenary,  y  =  -\e<=  +  e    f.  --^  = -\e'^  -  e   <=}. 


24.  The  tractrix,  x  =  a  log  ~-^ Va 


y 

(ly  y 


dx  ^a'  - 


Find  the  following; 

«/    \x        X  J  x"    '    3a;'    ' 


^  3(a;^  +  2)^ 
27.  f{%x^  -  ^x^  +  2.-6~'  +  i«"^)d^.  ^"^  ~  ^^   4-  (? 

2Q.J-—-dx.  ^^+^' 

29.    f—^—^.  —^^^j^a. 

^   {a-  —  x"y  a'Va'  —  '^^ 


30.    f-— 

"^  xV2 


X 

dx  ^'2ax—x^ 


V2ax  -  x"  ax 


C, 


^^-/!-qf^-  log(.^  +  2.  +  3)  +  C'. 

^^-  /^^'^^-  ^  -  3  log  (2a:  +  3)  +  a 

35.  f—J^^.  i  log  (2a;  +  4/rf4^)  +  ^. 

36.  f  ^^    log  (a; - 2 + Vx' - 4a;+ 13)  +  (7. 

'^     VX'  -  4:X  -\-   U  ^ 


GENERAL  DIFFERENTIATION,  81 

38.  /     ,  sin  ^  — 7= — h  (7. 

^    4/1  +  3a;  -  x'  Vl3 

^r.      r          dx  1           ,  a;  —  3   ,    ^ 

39.  /  -^ — — -.  -^  tan-i ^^  +  G. 

^  x^  —  Qx  -{-11  y-                 |/;> 

40.  y   (:c'-2a;+2)(a;-l)(?a;.  -  f. 

/3a;c?a;  ^  ,-— 

., 1/125. 
2 

— r^.  2f  -  log  3. 

0 

43.  /  3  sin''  x  cos  xdx.  sin'  a;  +  C. 

44.  /  3i(a  —  J  cos''  a-)^  sin  x  cos  a;^a-.         [a  —  b  cos'  a;)^-f^' 

45.  f^^^AJidx.  log(tanx  +  x)-\-C. 

^    tan  a;  +  .r  ^  ^            '     ^  ' 

46.  /  (tan  x  -\-  cot  xydx.  tan  a-  —  cot  a;  +  C. 


^''  di  ""  ^     ~       ''^  +  C  =  0.  y^^    _    ^y- 

48.  Given  ?/   —  2«a;?/  A-  x'  =  <^,  to  find  -^.  ^ ■ — r^- 

^  -^  dx"  {y  —  ax) 

49.  In  yd'^y  +  ^i/^  +  ^^^  =  0,  change  the  independent  vari- 
able from  X  to  y.  d^x      dx^      dx  _ 

dy'^      dy^      dy  ~ 

50.  Change    the    independent   variable    from    a;    to    «    in 

(2a;  -  1)^^-1  +  (2a;  -  1)^^  =  2y,  where  2a;  =  1  +  e^ 

51.  If  ?/'  =  dec  2a;,  prove  that  -~  =  3y^  —  y. 


82  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

53.  Griven  s  =  f  —  5f  +  6^;  find  the  velocity  (v)  and  its 
rate  of  change  {a')  when  t  =  10.  v  =  206;  a'  =  50. 

53.  In  ^  seconds  after  a  body  leaves  a  certain  point  the  rate 
of  change  of  its  velocity  is  6t  —  12  feet  per  second;  required  its 
velocity  and  distance  from  the  point  of  starting. 

V  =  3f  -  12t;  s  =  e  -  6t\ 

54.  In  the  last  example  how  far  will  the  body  travel  in  10 
seconds  ?  _  ^^■f^_^  ^^f^^  3.3  ^  433  ^  ^g^  ^^^_y 

55.  How  many  times  faster  is  x  increasing  than  log  x,  when 
X  =  n?  )i  times. 

56.  Required  the  value  of  x  at  tlie  point  where  the  slope  of 
the  curve  y  =  tan  x  is  2.  tt 

T* 

57.  A  man  is  walking  on  a  straight  path  at  the  rate  of  5  ft. 
per  second;  how  fast  is  he  approaching  a  point  120  ft.  from  the 
path  in  a  perpendicular,  when  he  is  50  ft.  from  the  foot  of  the 
perpendicular  ?  ,  l^f  ft.  per  sec. 

58.  A  vertical  wheel  whose  circumference  is  20  ft.  makes  5 
revolutions  a  second  about  a  fixed  axis.  How  fast  is  a  point 
in  its  circumference  moving  horizontally,  when  it  is  30°  from 
either  extremity  of  the  horizontal  diameter  ?  50  ft.  per  sec. 

59.  A  buggy  wheel  whose  radius  is  r  rolls  along  a  horizontal 

path  with  a  velocity  v';  required  the  velocity  (-^j  of  any  point 
{x,  y)  in  its  circumference;  also  the  velocity  of  the  point  hori- 
zontally f-^j  and  vertically  (-^). 

The  curve  described  by  the  point  in  the  circumference  of  the 
wheel  is  a  cycloid  whose  equation  is  ^  =  r  vers~^  -  ~  V^ry—y"^] 
differentiating  this  and  dividing  by  dt,  we  have 


dx  _  y         dy 

dt  ~  J^iry  —  «/^  ^^ 


(1) 


GENERAL  DIFFERENTIATION.  83 

Again,  the  abscissa  of  the  center  of  the  wheel  is  r  vers"^  -: 

r 

differentiating  this  and  dividing  by  dt,  and  we  have 

V2ry  —  y^'^i^ 
Again,  since  ds^  =  dx^  -f  dy'^,  we  have 


|=^(|)V(f)" (3) 

From  (1),  (2),  and  (3)  we  readily  obtain 

60.  In  the  last  example  find  the  values  of  -r— ,  -~,  and  -7-  at 

dt    dt  dt 

the  point  (1)  where  y  =  0;  (2)  where  y  =  r\  (3)  where  y  =  2?-. 

61.  Water  is  poured  at  a  uniform  rate  into  a  conical  glass  3 
inches  in  height,  filling  the  glass  in  8  seconds.  At  what  rate  is 
the  surface  rising  (1)  at  the  end  of  1  second?  (2)  At  what  rate 
when  the  surface  reaches  the  brim  ? 

(1)  \  in.  per  sec.     (2)  \  in.  per  sec. 


CHAPTER  Y. 

SERIES,   DEVELOPMENT  OF  FUNCTIONS,  AND 
INDETERMINATE   FORMS. 

SERIES. 

113.  A  Series  is  a  succession  of  terms  following  one  another 
according  to  some  fixed  law. 

If  the  sum  of  the  first  n  terms  of  an  infinite  series  approaches 
a  definite  limit  as  n  increases  indefinitely,  the  series  is  Conver- 
gent ;  if  not,  it  is  Divergent. 

The  sum  of  a  finite  series  is  the  sum  of  all  its  terms;  and  the 
sum  of  an  infinite  convergent  series  is  the  limit  which  the  sum 
of  the  first  n  terms  approaches  as  n  increases.  An  infinite 
divergent  series  has  no  definite  sum. 

114.  To  Develop  a  function  is  to  find  a  series,  the  sum  of 
which  shall  bo  equal  to  the  function.  Hence  the  development 
of  a  function  is  either  a  finite  or  an  infinite  convergent  series. 

For  example,     (a;  +  1)'  =  a;^  +  3.r^  +  3a;  +  1. 

This  finite  series  is  the  development  of  the  function  {x  +  1)^ 
for  any  value  of  x. 

Again,  by  division  we  obtain 

-^=l  +  x  +  x'' -\-x'  +  ...  x^^-K    .     .     .     (1) 


Now  this  series  is  the  development  of  = only  for  values 

J.  —  tC 


84 


SERIES.  85 

of  X  numerically  less  than   1,  for   the   omitted  remainder  is 


X" 


-;  hence,  denoting  the  series  by  s,  we  have 


1-x- 

1      _  a:" 

1  _  a;  -  ^  +  nr^- 

1  X"' 

Therefore  s  can  be  the  value  of only  when =  0 

1  —  X         -^  1  —X         ' 

and  s  the  development  of  :j only  for  such  values  of  x  as  will 

x" 
cause to  approach  0,  as  n  increases  indefinitely,  and  this 

can  be  the  case  only  when  a;  <  1. 
Thus:  (1)  For  a;  =  3  we  have 

-1=1  +  2  +  4  +  8  +  ...  2"-i  -  2", 

which  would  be  absurd  were  the  remainder  —  2"  omitted. 
(2)  For  a;  =  ^  we  have 

^       2=l  +  l  +  ^  +  i+.      J-  +  i 

in  which  the  remainder  —  decreases  as  n  increases,  indefinitely. 

115.  A  series  is  said  to  be  absolutely  convergent  when  it 
remains  convergent  on  making  the  signs  of  all  its  terms  positive; 
but  only  conditionally  convergent  when  it  becomes  divergent  on 
such  a  change  of  signs. 

The  series  1  —  i  -|-  3  —  i  +  .  .  .  is  an  example  of  a  condi- 
tionally convergent  series. 

1 16.  Prop.  TJie  inpiite  series u^-{-  u^-\- ...  u„_i  +  w„  +  . . . 
will  be  absolutely  convergent  if  the  terms  are  all  finite,  and  the  limit 

of  the  ratio  — ^ ,  as  n  is  indefinitely  increased,  is  numerically  less 

tha7i  unity. 


86  DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

qt 

Since  the  limit  of  — -  -,  as  n  is  indefinitely  increased,  is  less  than 

1,  there  must  exist  some  finite  integer,  c,  such  that  for  all  values 

u 
of  11  which  are  greater  than  c,  — ^  is  less  than  1. 

The  sum  of  the  terms  u^ -\- u^-{-  .  .  .  ?<<,  is  a  definite  finite 
quantity,  and  it  only  remains  to  show  that  the  sum  of  the  remain- 
ing terms,  w^+i  +  u^^^  +  .  .  .,  is  also. 

Let  r  be  a  number  less  than  1  but  greater  than  any  of  the 

ratios  '^,  ^'^,,.,;  then 

Uc+3  <  r  Mc+2  <  r'  u^+i  <  r'  u^, 


iic+i  +  Uc+2  +  Uc+3  +  .  .  .  <  ?/c(l  -\- r -\- 7'^ -{-  .  .  .)r, 
or        w<,+i  +  ?^„+2 -\-u,^s-^  .  .  .  <r Wc(^^-^3^,j, 

the  last  member  of  which  is  finite,  since  r  <  1,  Art.  114;  there- 
fore the  first  member  is  also  finite. 

Again,  that  the  sum  u^+i  +  '^^c+2  +  •  •  •  >  though  finite,  may  be 
indeterminate,  is  precluded  by  the  fact  that  the  limit  of  u„,  as  n 
approaches  oo ,  is  0.     Therefore  the  series  is  convergent. 

Scholium.  The  limit  of  u„  as  n  increases  is  always  0  in  case 
of  a  convergent  series,  but  the  mere  fact  that  the  limit  of  its 
nth  term  is  0  does  not  prove  a  series  to  be  convergent. 

117.  Cor.  I.  The  series  a^  -\-  a^x  -\-  a^x"  +  .  .  .  fl'„_ia;""^ 
+  rt„a;"  +  . . . ,  where  «„,  a,,  a^,  etc.,  are  independent  of  x,  is  con- 
vergent for  all  values  of  x  numerically  less  than  k  (say),  the  limit 

of  -^=i,  as  n  approaches  oo  . 

For,  by  Art.  116,  the   series   is   convergent  if  the  limit  of 


SERIES.  ■     87 

a„x^  -^  a-n^iX"'^,  or  a„x  -^  a^.^,  is  less   than   1,  and  therefore  if 
X  <  h. 

118.  CoK.  II.  When  x  >  Ic  the  series  is  divergent,  and  when 
X  =  k  the  series  in  some  cases  is  convergent,  and  in  others  di- 
vergent. 

119.  Cor.  III.  The  series  f\x)  =  «,  +  la^x  +  ^.a^x""  +  .  .  . 

{n  —  l)an-iX"~^  -\-  na^x"'^,  obtained  by  differentiating  f{x)  =a^ 

+  a^x  -\-  a^x^  +  .  .  •  ««-ia;"~^  +  «„a;%  is  convergent  for  the  same 

values  of  x  as  the  last-mentioned  series. 

/^ 1^/)^ 

For  in  the  former  Tc  =  the  limit  of  ^^ ^_±=1  which  is  the 

same  as  the  limit  of  — ^ . 

It  is  also  evident  that  the  limits  of  convergence  of  the  series 
obtained  by  integrating  the  individual  terms  of f{x)dx  are  the 
same  as  those  of  the  series  f{x)  itself. 

EXAMPLES. 
Find  the  values  of  x  which  will  render  the  following  coi>- 
vergent : 

°  2  1  n-1  n 

1.1+.+^+-+...^—;+-+... 

Here  a^-i  = r  and  a„  =  -.     .-.  ■ —  = =  1  H :, , 

n—\  n  a„       7i  —  1  n  —  1 

which  =  1  when  w  =co.     Hence  (117),  —  1  <  .-c  <  1;   that  is, 

X  lies  between  —  1  and  +  1' 

B-l 

13     '    14    ■  ■  '   |w  —  1     '    |?i- 


/y^  /y  /y^  «"— 1  /«" 


Here  — —  =  -; — =^-  =  n,  which  =  oo  when  w  =  oo  :  hence 
«„          \n  —  1  ' 


—  CO  <  a;  <co  ;  that  is,  the  series  is  convergent  for  all  finite 
values  of  a;. 

3^      ^x"-      7^'  2(y^-l)+l      ^  ,  2y^  + 1 

^-    3  +    5+10   +  •  •  •  +  (^_i)'+r'*^"    +  ^^  +  1^  +  •  •  • 


DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

Here  —  =  r ,  ,„   ,   ,  X  ;, — r— r?  which  =  1  when  w  =  oo . 

a„        {n  —  1)'  +  1      2n  +  1 

.-.  —  1  <  a;  <  1. 

4.  a;  +  2':r'  +  3V  +  .  .  .  (^i  -  1) V'^  +  wV  H-  .  •  . 

-  1  <  a;  <  1. 

5.  ^       +       '^^-  +  ...         ^""^  +  — ^^+... 
1  +  i/l       1  +  4/v!     '  1  +  4/^i_i       1  -^  |/^i 


-  1  <  «  <  1. 


DEVELOPMENT   OF   FUNCTIONS. 


120.  There  are  two  conimou  and  useful  formulas  for  de- 
veloping functions :  Taylor's  and  Maclaurin's. 

We  shall  deduce  Taylor's  formula  first,  as  Maclaurin's  may 
be  derived  from  it. 

121.  Taylor's  Formula  is  a  formula  for  developing /(^  -f  x) 
in  a  series,  where  f{y  +  •'c)  represents  any  function  of  the  sum  of 
two  variables,  such  as  (?/  +  ^y\  log  {y  +  ^)>  sin  {y  +  x),  a^+^. 

The  derivation  of  Taylor's  formula  may  be  regarded  simply 
as  the  process  of  finding  the  acceleration  of  a  function.  Thus, 
let  u  =f{y),  and  suppose  y  to  be  increased  by  x,  we  then  have 
(Art.  24), " 

^w  =f{y  +  x)  -f{y)  =  Ax  +  m,x',     .     .     (1) 

or  f{y +  x)  =f{y) -^Ax  +  m,x', (2) 

where  A  {=  m^  =  f'{y) )  is  a  constant  with  respect  to  x,  and 
fn^x'  is  the  acceleration  of  ?/  (Art.  25),  and  this  is  what  we  wish 
now  to  determine. 

Since  m„  is  of  such  a  character  that  m^x  vanishes  with  x,  we 
will  assume  (see  footnote,  p.  10) 

m,  =  B+  Cx  -\-Dx'-{-...  Lx"'',    ....     (3) 

where  B,  C,  D,  .  .  .  L  are  independent  of  x,  and  x  has  such  a 
value  as  to  render  the  series  (if  infinite)  convergent.  Substitut- 
ing in  (2),  we  have 

f{y  +  ^)  =  f{y)  +  ^^  +  Bx^  +  (7a;=  +  .  .  .  Z.t«-\     .     (4) 


DEVELOPMENT  OF  FUNCTIONS.  89 

Differentiating  (4)  successively  with  respect  to  x,  we  have 
f'{y  +  x)     =    A-\-2Bx-\-Wx''  +  .,.{:u-l)Lx''-";     (5) 
f'\y^x)    =2B  +  QCx  + ..  .{n-l){n-2)Lx''-^;      (6) 
f"'{y  +  a;)   =  6(7  +  ...(«  -  1)(>^  -  2)(^^  -  3)Z:r"-^;      (7) 

5 

f--\y  +  ^)  =  \>i^L (8) 

These  equations,  (5),  (6),  (7),  etc.,  are  true  for  any  value  of 
jc  which  renders  equation  (4)  convergent  (Art.  119);  therefore 
they  are  true  when  x  =  0,  which  gives 

f'{y)     =    A,  .-.  A=f'{yy, 

r\y)    =2B,  ,'.B  =  ~r{y); 

f"'{y)  =6  0,  .-.  C=~f'''{y); 


Substituting  these  values  for  A,  B,  C,  .  .  .  L  in  (4),  we  have 

Ay+^)  =f{y)  +f'{y)^  +  riy)~  +/'"(^){^         ] 

«-i        y  (A) 

This  is  the  formula  required,  which  was  first  published  in 
1715  by  Dr.  Brook  Taylor,  from  whom  it  takes  its  name. 

The  preceding  is  not  a  rigorous  demonstration  of  Taylor's 
formula,  inasmuch  as  the  possibility  of  development  in  the 
proposed  form  is  assumed.  A  rigorous  proof,  including  the 
form  of  the  remainder,  has  been  inserted  in  the  Appendix,  A. , 
to  be  used  or  not,  according  as  the  teacher  or  student  may 
desire. 


90  DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

122.  Cor.  I.  To  determine  for  what  values  of  x  the  series 
is  convergent. 

The  wth  and  {n  -\-  l)th  terms  of  the  series  are  evidently 


r-'iy)wr^  ^^^  r(y) 


X 

\n 


Therefore  (Art,  117)  the  series  is  convergent  for  any  vakie 
of  X  numerically  less  than 

r-Ky)   ^rM^     ^^    J=^m  when  ..=... 
\n-l  \n  \  fiy)  1' 

Hence,  if  •  ^  /";     is  not  zero  when  n  =cc ,  the  series  is  con- 

^"*"  .    r-'M 

vergent   for  all  finite  values  of  x;  and  if  n—^.-^  =  0,  when 

n  =^0 ,  the  series  is  divergent  for  all  values  of  x  except  0. 

In  deducing  Taylor's  formula  we  have  supposed  all  the  func- 
tions that  occur  to  be  continuous.  Hence  the  formula  is  inap- 
plicable, or  "fails,"  if  the  function,  or  any  of  its  differential 
coefficients,  be  infinite  for  values  of  the  variable  lying  between 
the  limits  for  which  the  development  holds. 

123.  To  develop  {y  -f  x)"'. 
Here  /(«/  -{- x)  =  {y  -\r  xY'K 

Make  x  =  0,  f{y)  =  y"". 

Differentiate,  etc.,    f'{y)  =my^~^, 

f"{y)  =  m{m  -  l)y'"-\ 
f"'{y)  =  m{m  —  l)(w  -  2)y'"-^, 
etc.  etc. 

Substituting  these  values  in  (A),  we  have 

{y  -\-  a;)™  =  y""  -\-  mx  «/™~^  -| ^^-y^ -x'y 


"i,,, 1111—2 


,  m(m  —  l)(w  —  2)   3  ^  ,     ,  .^, 

+  — ^^ -^ — —^x'y'^-^,  etc.,      (B) 

11    • 
which  is  the  Binomial  Theorem. 


DEVELOPMENT  OF  FUNCTIONS.  91 

Cor.  I.  Let  us  determine  for  what  values  of  x  the  equation 
is  true,  supposing  m  negative  or  fractional. 

/«-i(y)  =  m{m  —  l)...{m-n-\-  2)/y™-"+S 

/"(?/)  =  m{m  —  1)  .  .  .  {m  -  n  +  1) «/""-"; 


.-.  (Art.  122), 


/"{y)        m  -n  +  r 


which  =  —  y  when  w  =  oo . 

Therefore  formula  (B)  is  true  when  x  is  numerically  less 
than  y. 

Cor.  II.  Making  y  in  (B)  equal  to  1,  we  have 

/I  ,    X,.,     -,  ,         ,    '^M^n  —  1)  „  ,  m(m  —  l)(m—2)  ,  ,     ^  ,^. 

{l-^x)^=l+mx-\-  —^ -V+-^ ^ -V  +  etc.,     (C) 

in  which  —  1  <  a;  <  +  1. 

124.  To  develop  sin  {y  +  x). 

Here  f{y  -\-  x)  =  sin  (y  +  x)- 

Making  x  =  0,  and  differentiating,  we  have 

f{y)  =  sin  y;         f'{y)^cosy;    f"{y)  =  -smy; 
f'"{y)  =  -cosy;    /'"{y)  =  smy;     etc. 
Substituting  these  values  in  (A),  we  have 


sin  {y-\-x)=smy{l-^+  -r^-y  +  etc.\ 

,  /         x'        x'       x\     ^   \  \ 

+  cos2/(a;-|3-+|^---+etc.jj 


r  (D) 


Cor.  I.  In  (D)  by  making  y  =  0,  remembering  that  sin  0  =  0 
and  cos  0  =  1,  we  have 

x^        x^        x^ 
sin  X  =  X  -  -r^-i-  —  -  -- -{-  etc.      .    .    .     (E) 

II        12.        \L 


92  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

CoE.  II.  Differentiating  (E),  we  have 

or  Q*  (T 

cosa;  =  l ---+---  —  + etc.       .    .     .     (F) 

Cor.  III.  For  the  quantities  within  the  parentheses  in  (D), 
substituting  their  values  from  (E)  and  (F),  we  have 

sin  [y  -\-  x)  ^=  sin  y  cos  x  +  cos  y  s,va.x.  .     .     .     (G) 

Cor.  IV.  Differentiating  (G),  regarding  x  as  constant  and  y 
as  variable,  we  have 

cos  (y  -{-  x)  =  cos  y  cos  x  —  sin  y  sin  a;.  .    .     .     (H) 

125.  To  develop  log  (y  -\-  x). 

f{y  -{-  x)  =  log  {y  -\-  x);  making  x  =  0,  and  differentiating, 
we  have 

/(2/)-log(y);  riy)  =  y,  r%)  =  -^; 
r'(y)  =  J.;       f'iy)  =  -  ^^  ^t«- 

Substituting  in  (A),  we  have 

log  (2/ +  a;)  =  log  (2/)+^- ^,  +  ^3 -etc.,    .    ,     (I) 

which  is  the  logarithmic  series. 

Cor.  I.  The  nih.  and  {n  +  i)th  terms  of  (I)  are,  omitting 

the  signs,  -. , ,   „  ,,  and  — -;  hence.  Art.  117, =  — ^-^, 

^       {n  —  l)y  ^^  ny"*  a„        n  ~1 

which  =  y  when  «  =  oo  .     Therefore  formula  (I)  is  true  when 

X  is  numerically  less  than  y. 

Cor.  II.  In  (I),  by  making  ?/  =  1,  we  have 

log(l  +  ^)=:.-^+y-^  +  etc.,     .     .     (K) 

which  is  true  for  all  values  of  x  numerically  less  than  1. 


DEVELOPMENT  OF  FUNCTIONS.  93 

126.  Maclaurin's  Formula  is  a  formula  for  developing  a 
function   of   a    single    variable,    as    y  =z  a^^    y  z=z  log  (1  +  x), 

y  =  {a  +  ^Y- 

It  may  be  derived  from  (A)  by  making  y  —  O,  which  gives 

f{x)  =  /-(O)  +/'(0)^  +/"(0)  ^  +  /'"(O)^ 


13 


iC' 


n-l 


+  --/-'(0)^^^  +  ...,    (L) 

in  which /(0),/'(0),/'"(0),  etc.,  represent  the  values  which /(.r) 
and  its  successive  derivatives  assume  when  a;  =  0. 

Cor.  I.  Formula  (L)  is  true  for  all  values  of  x  numerically 

less  than     -^  ,  when  71  =  qo  . 

127.  To  develop  a"". 

Here  f{x)  =  a%  .-./(O)      =  a'  =  l; 

f{x)=a=^loga,  .-./'(O)     =loga; 

r{x)=a^\og'a,  .-.  r(0)    =log^a; 

f"{x)  =  a-  log'  a,  .'.  /'"(O)  =  log'  a; 
etc.  etc. 

Substituting  these  values  in  (L),  we  have 

x^  x^  X* 

a^  =  1  +  log  ax  +  log^  a-r^  +  log"  a-^  +  log*  a-r^,  .     (M) 

which  is  called  the  Exponential  Series. 

Cor.  I.  This  series  is  convergent  for  all  finite  values  of  x, 
since /"~X^)  -^/"(O)  is  obviously  finite  and  different  from  zero 
for  all  values  of  n. 

Cor.  II.  Making  a  =  e,  remembering  that  log  e  =  1^  Ave 
have 

e-  =  i  +  ^  +  ^  +  -^  +  ^  +  etc..    ,    .     (N) 


94  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

CoE.  III.  Making  x  =  I,  we  have 

^-l  +  l  +  ^+^+-^+4+etc.   .     .     (P) 

Hence  e  =  2.718381  +. 

128.  Find  the  development  of  tan" ^  a;. 

In  the  applications  of  Maclaurin's  formula  the  labor  of  find- 
ing the  successive  derivatives  can  often  be  lessened  by  taking 
the  development  of  the  first  derivative,  as  follows : 

f{x)  =  tan  -'x,  .:  /(O)  =  tan  -  ^  0  =  0 ; 

f{x)  =  -^,  =  l-x^  +  x'-x'  +  etc.,     .-./'(O)    =1: 

1   -f-  X 

f"{x)  =:  -  2a;  +  4a;'  -  Qx'  +  etc.,  .'./'(O)    =  0; 

f"{x)  =  -  2  +  3  .  4  .  a;=  -  5  .  6a;^  +  etc.,  .-.  /"'(O)  =  -  2; 

f^{x)  =  2 .  3  .  4a;  -  4  .  5  .  6a;'  +  etc.,  .-.  /^(O)    =  Oj 

f{x)  =  |4  -  3  .  4  .  5  .  6a;^  +  etc.,  .'.  /"(O)     =  |4 ; 

f^{x)  =  -2. 3. 4. 5.  6a;  +  etc.,  .'.  /'^(O)    =  0; 

f^{x)  =  -  |6_+  etc.,  .-. rm  =  -  |6; 
etc.,  etc. 

Substituting  in  (L),  we  have 

(T^  /v^  /v' 

tan-^a;  =  a;— — +  - —  +  etc.    .     ,     .     (Q) 

o  0  i 

EXAMPLES. 
Develop  the  following: 

/y»3  /yy^  ry>^  ^'7' 

vt/  ^  %0  %JfAJ 


1.  «/!  +  .'.  i  +  ___  +  ___+eto. 

Put  x^  =  y,  and  develop;  then  replace  y  by  its  value. 

2.  {a-\-x)-\  a-3  -3a-^a;  +  6aV— lOa-V  +  etc. 


DEVELOPMENT  OF  FUNCTIONS.  95 

,   a;^         X*         x"  x^       ,     , 


K        pCOSX 


^(1-   3+^4--  -j6-  + etc. j. 


6.  tan  a;.  a;  +  —  +  — ^  +  etc. 

7.  sec  a;.  "*  ^~  Y  ^  ^  "^  ^*°' 

'/'^  '7'^  ^* 

8.  log  (1  +  sin  a;).         ^  ~  y  +  "e  ""  12  "^  ^**^ 

9.  log  (1  +  e%  log  2  +  I  +  ^  -  etc. 

x" 

10.  e^si°^.  1  +  ;y=  ^  •    ^  etc. 


2x^ 

11.  e^  sec  x.  l-\-x-{-  x''  -\-  —  -{-  etc. 

o 

12.  log  (1  -  a;  +  a.-).       -  a;  +  |- +  ^+  j  -  etc. 


Put  —  a;  +  a;'  =  ^,  and  develop;  then  replace  y  by  its  value. 
In  the  two  following  examples  put  y  for  x\  develop,  and  re- 
place y  by  its  value. 

13.    fl-».-.  ^i_-^____.__-eto. 

14        1  i_5:  +  ^_li££l  +  etc. 

yl  +  a;''  -^       '^■^       -v.^t.o 

129.  To  find  the  value  of  n. 
We  find  by  development  that 

sm-a;  ^x  +  --  +  ^-^^  +  ^-^-^-^  +  etc., 

where  x  lies  between  —  1  and  +  1. 


96  DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

7t 

Making  a?  =  i,  remembering  that  sin"^  -J  =  — ,  we  have 

^^ifi  +  l.A.  i  _^+^ 

6       3  \   ^  24  ^  640  ^  7168  ^j' 
or  7t  =  3.141592  +  .  .  . 

130.  To  compute  Natural  logarithms. 

We  have  found 

\og{l  +  x)=x-\+\-\  +  \-^tc.,.     .     (1) 

where  x  is  numerically  less  than  1. 

We  now  proceed  to  modify  this  series  so  that  it  shall  be  true 
and  convergent  for  larger  values  of  x. 

Substituting  —  x  for  x  in  (1),  we  have 

log  (1  -  a;)  =  -  a;  -  ^  -  |-  -  ^  -  |-  -  etc.    .     (2) 

Subtracting  (2)  from  (1),  Ave  have 
log(l  +  x)-log(l-a;)  =  2(a;  +  ^+^+|'  +  etc.).     .     (3) 

In  (3)  make  x  =  ,  where   z   may  have   any  positive 

value;  then  r = ■ 

^  1  —  X  z 

and        log  (1  +  a;)  —  log  {I  —  x)  =  log  (2  +  1)  —  log  (5;), 

we  have       log  {z-{-l)  =  log  {z) 

I  1  ~l 

(4) 


+  3 


+  ^7^7:rr-rT^  +  etc. 


L22  +  1   '     3(22  +  1)'   '    5(22  +  1)= 
By  this  series  we  can  compute  the  Natural  logarithm  of  any 

number  (2  +  1)  when  we  know  the  logarithm  of  the  number  {z) 

less  by  unity. 

Making  z  =  1,  remembering  that  log  (1)  =  0,  we  have 


log  (2)  =  2 


-  -1-  -^  +  -^  +  ^^  +  etc.  I. 
3^3.3=^5.3^^7.3'^        J 


DEVELOPMENT  OF  FUNCTIONS,  97 

Taking  six  terms  of  this  series,  we  have 
log  2  =  .693147+  .  .  . 
Putting  2;  =  3  in  (4),  we  have 

log  (3)  =  log  3  +  2(i  +  J3,  +  J^.  +  ^,  +  etc.) 

=  1.098612  + 
log  (4)  =  2  log  2  =  1.386294  +. 

Putting  z  —  4,,  we  have 

log  (5)  =  log 4  +  2(^  +  3^  +  ^-1^.  +  ^-L,  +  etc.) 

=  1.609437  +. 
log  10   =  log  2  +  log  5  =  2.302585  +. 

In  this  way  we  can  compute  the  natural  logarithms  of  all 
numbers.  It  is  not  necessary  to  use  the  formula  in  finding  the 
logaritlims  of  composite  members,  for  they  can  be  found  by 
simply  adding  the  logarithms  of  their  factors.  Thus  log  15  = 
log  3  +  log  5. 

131.  To  compute  common  logarithms. 

The  modulus  of  the  common  system  is  m  =  logj„  e  (Art. 
79).     Hence   10™  =  e,  .'.  log   (10'")  =  log  e,   or  m  log  10  =  1. 

■•■'"=  11^  =  2-!ol58B  =  -«^^^*+- 

Let  logj„  V  =  n    and    log  v  =  n'\ 

then  10"  =  V    and         e"'  =  v; 

10"  =  e',      log  (10")  =  log  e',  or 
n  log  10  =  n\  .'.  n  =  {n')  .434294  +. 

Hence,  to  find  the  common  logarithm  of  any  number  we 
multiply  the  natural  logarithm  of  that  number  by  the  modulus 
of  the  common  system. 


98  BIFFEBENTIAL  AND  INTEGRAL  CALCULUS. 


INDETERMINATE   FORMS. 

132.  In  algebra  —  is  called  a  symbol  of  indetermination, 
since  any  number  whatever  may  assume  this  form. 

Thus,  w  X  0  =  0 :  divide  both  sides  by  0  and  we  have  n  —  -. 

There  are  many  fractions  which  assume  the  form  of  —  in 

consequence  of  one  and  the  same  supposition,  which  makes  both 
numerator  and  denominator  =  0.  Such  fractions  are  called 
Vanishing  Fractions,  and  their  values,  which  appear  under  the 

form  of  — ,  can  generally  be  determined  by  the  calculus. 

x^  —  a'  0 

Thus  the  fraction  —. ;  becomes  —  when  x  =  a. 

x"  —  a"  0 

This  form  arises  from  the  existence  of  a  factor  {x  —  a)  com- 
mon to  both  numerator  and  denominator,  which  factor  becomes 
0  under  the  particular  supposition.  Dividing  both  terms  by 
this  factor,  we  have 

x^  —  a^      x""  4-  ax  4-  a'      ,  .  ,         Sa     , 

-,  which  =  -—  when  x  =  a. 


x^  —  a^  X  -\-  a  2 

133.  To  evaluate  a  fraction  that  takes  the  form  of - 

Let  u  and  v  be  functions  of  x  such  that  when  x  =  a,  ti  =  0, 
and  V  =  0. 

Let  u  and  v  be  estimated  from  the  point  where  their  values 
are  0,  that  is,  from  where  x  =  a;  then  when.'r  (=  a)  is  increased 
by  h  we  shall  have,  identically, 

u  _  Au 
V   ~  Av 


INDETERMINATE  FORMS 


99 


As  li  approaches  0,  or  x  approaches  a,  the  limit  of  --r-  is 
du 


equal  to  -^  (Art.  27) ;  therefore 


-1  = 


du' 
dv 


which  is  read,  when  x  =  a,  —  is  equal  to  -,— . 

V  dv 

Applying  this  to  the  preceding  example,  we  have 

x'  -  an      _  d{x'  -  a°)~l    _  3^'~ 
^  d{x'-  d')\^-2xl 


3a 


X   —  a 

An  easy  deduction  of  the  rule  for  reckoning   forms  like 

—  is  obtained  by  the  use  of  Taylor's  formula,  as  follows: 

Let  (p{x)  and/(.r)  be  two  functions  of  x  such  that /(a;)  =  0 

(p{a)  _  0 
0' 


and  0(a;)  =  0,  when  x  =  a,  then  we  shall  have 


/(«) 


Evidently 


0(0) 


limit 
h  =  0 


'd){a  +  7i) 


] 


limit 
7i  =  0 


■0(a)  +  0'(«)7H-|0'>)A^.  .  n  _  (p'{a)  _  (t>{x)- 


.n      0^)  _ 

L/(«)  +f'{a)li  +  WiaW  +  .  .  J  ~  /'(«)  "  /(^)J  ; 
Cor.  I.     If  (p'{a)  =  0  and/'(«)  =  0,  we  obviously  have 

(p''{a)  _  (p(x)l 
/"{a)  -  f{x)]^  ' 

and  if  0"(a)  =  0,  and /"(a)  =  0,  we  have 


(f)'"{a)  _  (p{x)' 
f'"{ct)  -  J{x)_ 


and  so  on. 


Hence,  Eule.  Substitute  for  the  numerator  and  denom- 
inator, respectively,  their  first  derivatives,  or  their  second 
derivatives,  and  so  on,  till  a  fraction  is  obtained  whose  terms  do 


100        DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

not  both  become  0  lulien  x  —  a  ;  the  values  thus  found  ivill  be  the 
true  value  of  the  vanishing  fraction. 

EXAMPLES. 

Compute  the  following: 
x'  -V 


1. 


X'  -  1 

a;^  -  16     n  8 


X    —  X 


3^+2    ~| 


7. 


X 

sm  x' 


-Jo 


X 

x  —  sm  X 


0; 
1; 


sm  X 


-  ^     log  sin  a;  ~| 
•  {7t  -  2xy], 


•z 


G. 


4 
sin   or  —   0"  f>ns  rrl 

13 


sm  X  —  x  cos  a;"! 


5^ 
2* 


^^  +  a;  _  20j;  9 


5a;'  -  8a;  +  3"]  2 

4 


.^        ^..   ,    3-1 

7a;^  -9a;  +  3j/  5* 

a;'  —  3a;  +  2    1  3 


2* 

^    1  —  cos  a;"]  >. 

0.  — ; .  u. 

sm^     Jo 

^    ff  -  4/a'  -  a;n  1 


X 

sin  3a;      1  3 

a;  —  f  sin  2a;Ja'  2* 

10.  'IJII^  .  .  3. 


1 


sec'  a;  —  2  tan  x~\  1 


1  +  COS  4a;     Jn-'  ^ 


a;  —  sm  x     J^ 

,/(a;-2)e'"  +  -^  +  2-|  1 

14.  ^,._.i)3        J^-    .  6- 


INDETERMINATE  FORMS.  101 

There  are  other  indeterminate  forms  besides  -,  such  as  — , 

0  CO 

CO  X, 0,  cx)  —CO,  0°,  CO  °,  1°°,  which  will  be  considered  in  succes- 
sion. 

134.  To  evaluate  a  fraction  that  takes  the  form  of  — . 

CO 

Let  u  and  v  be  functions  of  x  such  that  u  =cc  and  y  =co 

when  X  =  a:   then  for  the  same  value  of  x,  -  =  0  and  -  =  0. 

u  V   ■ 

Hence 

u      V       0      , 

—  =  —  =  -,  when  X  =  a. 

V       1_      0 

u 

'I 


u  \v  J       u  dv 


.'.  Art.  133,  —  =  — 7T-r  =  ~^ri-,  when  x  =  a.   .    .     .     (1) 

V        Jl\       v'du  ^  ' 

d[ 


Dividing  (1)  by  — ,  we  obtain 


udv  u 

1  =  —7-5     or     - 


du 
dv 


(2) 

Now  (2)  is  derived  from  (1)  by  dividing  by  — ;  hence,  if  —  is 
finite,  (2)  is  true  for  all  finite  values  of  — :  and  if  -  =  0  or  co  ,  it 

V  V 

may  be  shown  that  (2)  is  true  in  these  cases  also. 

11 
Suppose   — =  0  when  x  =  a,  and  h  a  finite  quantity, 

then  '±^k^'^±^  =  lc. 

V  V 

To  this  last  fraction  (2)  evidently  applies,  hence 

u  +  kv       du  4-  hdv  u    ,    ^       du    ,   ^ 

— = ^ ,     or    -  +  ^-  =  y-  +  ^; 

V  dv  V  dv 

,  1    ,  .  u      du         , 

that  IS,  —  =  -^— ,     when  x  =  a. 

V       dv 


102        DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 


If  -  —  CO ,  then  -  =  0,  and  we  have  the  preeeding  case. 
V  u 

Therefore  the  form  —  is  to  be  evaluated  in  the  same  way  as 

CO 


the  form 


EXAMPLES. 


Find  the  values  of  the  following: 

1     .-K   n 


log  X  _\^ 
il{x) 


2. 

3. 

4. 
5. 


d  (log  x)_ 

ax:"  +  d 

ex'  +  d 


]. 


X 


tan  X 

log  x~ 
cot  X 


„   log  cot  x' 
cosec  x  _ 
log  tan  2x 
los:  tan  X 


] 


tani  -(a;  +  l)j 


tan 


7tX 


log  cos  (^Ttx)  ~| 

log  (1  -  X)  J; 


=  X 


0. 


0. 
0. 

-1. 

2. 

1. 


INDETERMINATE  FORMS. 


103 


135.  To  evaluate  a  function  that  takes  the  form  of  0  X  oo 

or  CO  —  CO  . 

Functions  of  this  kind  can  be  transformed  so  as  to  assume 

the  form  of  -  or  — ,  and  then  be  evaluated  by  the   previous 
0        00  J  i. 

methods. 


EXAMPLES. 


Find  the  following: 
1.  sec  3a;  cos  bx\ 

This  takes  the  form  of  co  x  0;  but  sec  3a;  cos  bx  = 


_5 
~~3' 

cos  bx 
cos  3a;' 


which  takes  the  form  of 


0 


2.  sec  X  —  tan  a;] 


This  takes  the  form  of  oo  —  co  ;  but  sec  x  —  tan  a; 

sin  X       1  —  sin  a;      n  .  ■,    ,   ,       , ,      »  n  0 

= ,  winch  takes  the  form  oi  -. 

cos  X  cos  X  0 

3.  _i LI  . 

log  X      •'^  —  1 J , 

4.  cosec^  X  —  — , 

X 

e  1 


cos  X 


&^  —  e      X  —  1 
6,  (1  —  tan  a;)  sec  2a;], 


r' 



a"  , 

7tX~\ 

'L 

a' 

—  tan 

2a  _ 

a~ 

8. 

X 

sm 

X_ 

• 

9.  a;"'  log"  x\.     {m  and  ?i  being  4--) 
10.  (1    •  x)  tan  iiTTx)]^. 


1 

2' 

1 

3' 

1 

2' 
1. 

4 


0. 

2 


104        DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

136.  To  evaluate  a  function  that  takes  the  foim  of  0°,  co  °, 
or  1". 

Take  the  logarithm  of  the  given  function,  which  will  as- 
sume the  form  of  0  X  co ,  and  can  be  evaluated  by  Art.  135. 
From  this  the  value  of  the  function  can  be  found. 

EXAMPLES. 
Eind  the  following: 


This  takes  the  form  of  1".     Making  y  =  f  1  +  -  |  ,  we  have 


is  found 


log  y  =  X  log  f  1  -i —  j.     The  value  of  x  log  f  1  -] —  j 

by  Art.  135  to  be  a.    Hence,  when  a;  =  co ,  log  y  =  a',  .*.?/  =  e". 

2.  ^x~\^,    or    x^\^.  1. 

-      1 

This  takes  the  form  of  go  °;  the  log  of  x^  is  -log a;,  the  value 

of  which,  when  a;  =  co ,  is  0 ;  hence  x'^A^  =  1. 

3.  (sin  xy^""  ^]^ .  1. 

4.  (cot  xY'""  ^]„.  1. 

5.  (e^  +  l)^J  .  e. 
e.  (tan  f  )•"?].  i 

7.  (cot  xY^^s^l.  -• 

1 

8.  t^log  {e  +  x)l.  '-' 

9.  Ve--^x\.  ^ 
10.  Vcos  2./]„.                    '                          7' 


INBETERMmATE  FORMS.  105 

11.  f  S5]  •  1. 


13.  (cos  mxY^_ 


-\nm^ 


137.  In  implicit  functions,  as /(a*,  ?/)  =  0,  the  derivative 

dv 

-~  can  be  evaluated  by  the  previous  methods  when  it  assumes 

an  indeterminate  form  for  particular  values  of  x  and  y. 

EXAMPLES. 

1.  Find  the  slope  of  x^  —  a'xy  -\-  F'y''  at  the  point  (0,  0). 

«y  aa;  cix         . 

"^y       a'  _  2¥^       a'  -  2b'-^ 
ax  ax 

thatis    ^--^_i^      or     2h'(^X        n'i^A  -  n'^y - 
^('•i'  3        072^/  \itx/  \ax  /  U/X 

a  ~~  io  -^ 
ax 

dy       .  a^ 

.'.  -f  =0    or    ^. 

dx  (T 

2.  Find  the  slopes  of  x^  —  Zaxy  +  «/'  =  0  at  (0,  0). 

0  and  CO , 

3.  Find  the  slopes  of  »*  +  ax^y  —  ay^  —  0  at  (0,  0). 

0  and  ±  1. 

4.  Find  the  slopes  of  y^  =  x{x  +  a)'  at  (—  a,  0).         ±  V—  a. 

5.  Find  the  slopes  of  x*  +  2ax''y  —  aif  =  Q  at  (0,  0). 

0  and  ±  V2. 


CHAPTER  VI. 
MAXIMA  AND  MINIMA. 

DEFINITIONS   AND  PRINCIPLES. 

138.  A  Maximum  Value  of  a  function  is  a  value  that  is 
greater  than  its  immediately  preceding  and  succeeding  values, 
and  a  Minimum  Value  is  one  that  is  less  than  its  immediately 
preceding  and  succeeding  values. 

Thus,  while  x  increases  continuously,  if /(^c)  increases  up  to 
a  certain  value,  sa,y  f{a),  and  then  decreases, /(«)  is  a  maximum 
value  of  /"(ft-) ;  and  if,  while  x  increases,  f{x)  decreases  to  a 
certain  value,  say /(6),  and  then  increases, /(J)  is  a  minimum 
value  of /(a:). 

For  example,  sin  x  increases  as  x  increases  till  the  latter 
reaches  90°,  after  which  sin  a;  decreases  as  x  goes  on  increasing; 
that  is,  sin  90°  is  a  maximum  value  of  sin  x. 

Again,  if  x  increases  continuously  from  0  to  5,  f{x)  =  x^  —  Qx 
+  10  will  decrease  until  x  becomes  3  and  then  it  will  increase; 
hence  /'(S)  =  1  is  a  minimum  value  off{x),  or  x''  —  6a;  +  10. 

Let  the  student  substitute  1,  2,  3,  .  .  .  10,  successively,  for  x 
infix)  =  a:'  —  18a;'  +  96a;  —  20,  and  thus  show  that  /(4)  is  a 
maximum  and/(8)  is  a  minimum. 

139.  Any  value  of  x  thai  renders  f{x)  a  maximum  or  a 
minimum,  is  a  root  of  the  equation  f  (x)  —  0  or  co  ,  if  f{x)  and 
f'{x)  vary  continuously  with  x. 

For,  if  we  conceive  x  as  always  increasing, /(a-)  changes  from 
an  increasing  to  a  decreasing  function  as  it  passes  through  a 

106 


MAXIMA  AND  MINIMA.  107 

maximum  value,  saj  f{a),  and  from  a  decreasing  to  an  increas- 
ing function  as  it  passes  through  a  minimum  value,  say  f{b). 
Consequently /'(a;)  must  change  sign  as  x  passes  through  a  or  b, 
Art.  25.  But /'(a;)  can  change  sign  only  by  passing  through  0 
or  CO .  Therefore  /'(«)  or  f'{b)  =  0  or  od  ;  that  is,  a  and  b  are 
roots*  otf'(x)  =  0  or  GO  , 

To    illustrate     the     preceding    principles    and    definitions 
graphically,  let  i/  =  f{x)  be  the  equation  of  the  curve  A77i;  then 
f'{x)  =  the  slope  of  the  curve  at 
the  point  P  or  {x,  y),  A\'i.  48.     As 
X  (=  OB)  increases,  the  point  P 
will  move  from  A  along  the  curve 
to  the  right,  and  y  or  f{x)  will  in- 
crease till  it  becomes  aa',  and  then        oaSb    c      d  e    h  vi  - 
decrease  till  it  becomes  bb',  and 

then  increase,  etc.  Therefore  rm/, ,  cc/,  ee'  are  maximum,  and  5^', 
dd'  are  minimum,  values  off{x).  The  slope  of  the  curve, /'(a;), 
is  evidently  positive  before,  and  negative  after,  each  maximum 
value  of  f{x);  and  negative  before,  and  positive  after,  each 
minimum  value  of /(a;).  Moreover,  at  the  points  where /(a;)  is 
a  maximum  or  a  minimum,  the  curve  is  either  parallel  or  per- 
pendicular to  the  axis  of  x,  and  therefore /'(;i')  =  0  or  co  , 

The  converse  of  this  theorem  is  not  always  true;  that  is,  any 
root  otf'{x)  =  0  or  CO  does  not  necessarily  render  f{x)  a  max- 
imum or  a  minimum.  It  is  our  purpose  now  to  determine 
which  of  the  roots  will  render  f{x)  a  maximum  and  which  a 
minimum. 

140.  If  the  sign  of  f'{x)  undergoes  no  change  as  f'{x)  passes 
through  0  or  co ,  the  corresponding  value  of  f{x)  will  be  neither 
a  maximum  nor  a  minimum. 

For,  so  long  as  the  sign  of  f'{x)  undergoes  no  change, /(a;) 


*  Here,  and  in  what  follows,  the  word  root  includes  the  real  values  of  x 
which  satisfy  the  equations /'(a*)  =  0  or  go  ,  whether/'(.T)  be  an  algebraic  or 
a  transcendental  function. 


108        DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

does  not  change  from  an  increasing  to  a  decreasing  function, 
nor  vice  versa. 

Cor.  I.  If  an  even  nnmber  of  the  roots  of  f'{x)  =  0  or  co 
are  equal  to  a,  then  x  =■  a  will  not  render  f{x)  a  maximum  Oi 
a  minimum. 

141.  If  tlie  sign  off'{x)  undergoes  a  cliange  as  f  {x)  passes 
through  0  or  cc ,  the  corresjjonding  value  of  f\x)  is  a  maximum 
or  a  minimum. 

For,  if  f'{x)  undergoes  a  change  of  sign,  f{x)  necessarily 
passes  from  an  increasing  to  a  decreasing  function,  or  vice  versa. 

Cor.  I.  If  an  odd  number  of  the  roots  of  f'{x)  =  0  or  co  are 
equal  to  «,  then  x  =  a  will  render  f{x)  a  maximum  or  a  minimum. 

Cor.  II.  Therefore,  omitting  the  equal  roots  of  which  the 
number  is  even,  every  real  root  of  f'{x)  =  0  or  co  will  render 
f{x)  either  a  maximum  or  a  minimum.  Now,  of  these  let  us 
find  which  will  render /(a;)  a  maximum  and  which  a  minimum. 

142.  Maxima  and  minima  of  a  fiLnction  occur  alternately. 
For,  suppose  that  f(a)  and  f{b)  are  maxima  of  f{x),  where 

a  <  b.  Just  after  passing  through /(r(),/(;t)  is  decreasing,  and 
increasing  just  before  it  reaches  /(J) ;  but  in  passing  from  a 
decreasing  to  an  increasing  state  it  must  pass  through  a  mini- 
mum; hence  there  is  one  minimum  between  every  two  consecu- 
tive maxima. 

Cor.  I.  Denote  tlie  roots  of/'(.T)  =  0  and/'(.r)  =  co  which 
will  render /(re)  a  maximum  or  a  minimum  by  a^,  a^,  a^,  a^, 
etc.,  in  ascending  order  of  algebraic  magnitude.  Then,  if /(a;) 
is  an  increasing  function  for  all  values  of  x  less  than  a^,  that  is, 
itf{a,  —  h)  is  positive,  h  being  ever  so  small,  /'(a J, /( (7.3),  /(a,), 
etc.,  are  maxima,  and/(aj,  /'(aj,  etc.,  are  minima;  and  if  f{x) 
is  a  decreasing  function  for  the  same  values  of  x,  that  is,  if 
f'{a,  —  h)  is  negative,  the  maxima  and  minima  will  be  inter- 
changed. 


MAXIMA  AND  MINIMA.  109 


RULES  FOR  FINDING  MAXIMUM  AND  MINIMUM  VALUES  OF 

FUNCTIONS. 

143.  The  preceding  principles  indicate  the  following  rule 
for  finding  the  values  of  x  which  will  render  any  function  as 
f{x)  a  maximum  or  minimum: 

Differentiate  the  function  f{x)',  make  f'{x)  =  0  and 
f'{x)  =  00  ;  find  the  real  roots  of  both  equations,  and  arrange 
all  of  them  in  order  of  algebraic  magnitude,  as  a^ ,  a.^ ,  a^ ,  etc., 
omitting  the  equal  roots  when  there  are  an  even  number  of 
thenij  substitute  -co  or  a^  —li,  h  being  very  small,  for  x  in 
f'{x),  and  (1)  if  the  result  is  +,  a^,  a^,  etc.,  will  each  render 
f{x)  a  maximum,  and  ffj  J  ^4  J  6tc.,  will  each  render /(.i")  a  min- 
imum; (2)  if  the  result  is  —,/(«j),/(«3),  etc.,  will  be  minima, 
a,ndf{a^),f{aj,  etc.,  will  be  maxima. 

144.  The  preceding  rule  requires  that  all  the  real  roots 
shall  be  found;  it  is  sometimes  desirable  to  know  independently 
whether  any  particular  root  as  a'  will  render  f{x)  a  maximum 
or  minimum.     This  may  be  done  thus: 

I.  Substitute  «'  —  7i  and  a'  +  A  for  x  in  f'{x),  7i  being  a 
small  quantity,  and  (1)  iff  {a'  —  li)  is  -{-,  and /'(a'  +  ^^)  is  —, 
f(a')  will  be  a  maximum;  (2)  \if'{a'  —  li)  is  —  and  f'{a'  +  h) 
is  -\-,f{a')  will  be  a  minimum,  and  \if'{a'  —  h)  and/'(a'  +  A) 
have  the  same  sign, /(a')  will  be  neither  a  maximum  nor  a  min- 
imum. 

145.  II.  Developing  f{x  —  h)  and  f{x  +  ^0  ^J  Taylor's 
formula,  substituting  a'  for  x,  transposing /(a'),  and  remember- 
ing that /'(«')  =  0,  we  have 

/(«'  -  h)  -f{a')  =/"K)|  -/'"(«')^  +/^(«')^  -      (1) 

\rj  \0  rx 

and 
f{a'  +  h)  -/(«')  =f'{a')^+r'{a')^+  f^(«')^  +.     (2) 


110       DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

If  li  be  taken  very  small,  the  sign  of  tlie  second  member  of 
either  (1)  or  (2)  will  be  the  same  as  the  sign  of  its  first  term. 
Hence,  iff"  {a')  is  negative, /(«')  is  greater  than  both /(a'  —  h) 
and  f{a'  -f-  h),  and  therefore  a  maximum;  while  if  f"{a')  is 
positive,  /(«')  is  less  than  both  f{a'  —  li)  and  f{a'  +  ^^)>  and 
therefore  a  minimum.  Iff"  {a')  =0,  and /'"(a')  is  not  0,/(a') 
is  neither  greater  tlian  both  f{a'  —  h)  and  f{a'  -|-  ''O  ^^^  ^^ss 
than  both,  and  is  therefore  neither  a  maximum  noj^*  a  minimum. 
lif"'{a')  as  well  as  f"{a')  is  0,  then,  as  before, /(«')  will  be  a 
maximum  or  a  minimum  according  as  f^^{a')  is  negative  or  po- 
sitive; and  so  on. 

Hence  if  a'  is  a  root  of /'(re)  =  0  or  co,  substitute  it  for  x  in 
the  successive  derivatives  of  f{x).  If  the  first  derivative  that 
does  not  reduce  to  0  is  of  an  odd  order, /(«')  is  neither  a  maxi- 
mum nor  a  minimum;  but  if  the  first  derivative  that  does  not 
reduce  to  0  is  of  an  even  order, /(«')  is  a  maximum  or  a  mini- 
mum, according  as  this  derivative  is  negative  or  positive. 

Note. — In  many  instances  this  rule  is  impracticable  on  ac- 
count of  the  great  labor  involved  in  finding  the  successive  de- 
rivatives. 

146.  The  following  principles  are  self-evident,  and  often 
serve  to  facilitate  the  solution  of  problems  in  maxima  and 
minima : 

(1)  If  c  is  positive, /(a;)  and  c  X  f{x)  are  maxima  or  minima 
for  the  same  value  of  x;  hence  a  constant  positive  factor  or 
divisor  may  be  rejected  in  finding  this  value  of  x. 

(3)  log /(re)  and /(a;)  are  maxima  or  minima  for  the  same 
value  of  x;  hence  log  may  be  rejected. 

(3)  c  -hf{x)  and  f{x)  are  maxima  or  minima  for  the  same 
value  of  x;  hence  the  constant  c  may  be  rejected. 

(4)  If  w  is  a  positive  Avhole   number,   [f{x)Y  and  f{x)  are 

p 

maxima  or  minima  for  the  same  value  of  x;  hence  in  [f{x)Y 

the  denominator  q  may  be  rejected,  or  in  yf{x)  the  radical  may 
be  removed. 


MAXIMA  AND  MINIMA.  Ill 


EXAMPLES. 


1.  Find  what  values  of  x  will  render  x^  —  ^x^  —  2^x  +  85  a 
maximum  or  a  minimum. 

Here  f{x)  =  x'  -  3x'  -  24:X  +  85,  f{x)  =  dx'  -  6x  -  24, 
f"{x)  =  6x  -  6.  Making /'(^)  =  0,  we  have  3a;'  -  6x  -  24  =  0, 
the  roots  of  which  are  x  =  -\-  4,  x  =  —  2.  Now  to  determine 
whether  these  values  of  x  give  maxima  or  minima  values  of 
f{x),  we  substitute  them  for  x  in/"(a;). 

Thus:  /■"(4)  =  6x4-6::^+ 18, 

f\-  2)  =  6  X  -  2  -  6  =  -  18. 

Hence,  Art.  145,  when  x  =  4, /(a:)  is  a  minimum,  and  when 
X  =  —  2,/(.r)  is  a  maximum. 

Let  the  student  construct  the  locus  of  y  =  x^—  3x^  —  24a;  -j-  85, 
and  thus  exhibit  these  results  graphically. 

2.  Examine  f{x)  =  x^  —  3a;'  +  3a;  +  7  for  maxima  and 
minima. 

Here /'(a:)  =  3a;=  -  6.f  +  3, /"(a;)  =  6a;  -  6,/'"(a;)  =  6.  The 
roots  of  3.a;*  —  6.r  +  3  =  0  are  a;  =  1,  a;  =  1.  Substituting  these 
values  of  x  \nf"{x)  and /'"(a;),  we  have /"(I)  =  0,/'"(l)  =  6. 
Therefore  the  f  unction/(a;)  has  neither  a  maximum  nor  minimum 
value,  which  we  also  infer  from  the  fact  that  the  two  roots  of 
f'{x)  —  0  are  equal.  Art.  140,  Cor.  I. 

3.  Find  the  maxima  and  minima  of  x^  —  5a;''  +  5a;'  —  1. 
Here  /'(a-)  ::=  bx'  -  2Qx'  +  15a;';    •'■/'{x)  =  0  is  5a;'  -  20a;'' 

-\-  15a;'  =  0,  or  (a;'  —  4a;  -J-  3)a;'  =  0;  the  four  roots  of  which  are 
0,  0,  1,  and  3.  Rejecting  the  two  equal  roots.  Art.  140,  we  have 
a^  =\,  a^—  3,  and  since  /'(—  co  )  =  5(—  c»  )'  is  +,  the  given 
function  is  a  maximum  when  a;  =  1,  and  a  minimnm  when 
X  =  3. 

Therefore  /(I)  =  0  is  a  maximum, 

and  /(3)  =  —  28  is  a  minimum. 


112        DIFFERENTIAL  AND  INTEGRAL   CALCULUS 

4.  Examine  {x  —  iy{x-\-  2)'"  for  maxima  and  minima. 
Differentiating  and  reducing,  we  have 

/'(«:)  =  (.T-l)'(:r+ 2)^(72; +  5) 

The  Foots  oif'{x)  ^  0  are  those  of  {x  -  1)'  =  0,  (a;  +  2V  =  0 
and  Ix  -\-b  =  0',  hence  there  are  three  roots  each  equal  to  1, 
two  each  equal  to  —  2,  and  one  equal  to  —  f. 

Eejecting  the  two  equal  roots,  we  have  a^  =  —  \  and  a^  =  1, 
and  since  /'(—  co)  =  (—  oo)^(—  co)^(—  7oo)  is  -\-,  f{x)  is  a  max 
imum  when  .t  =  —  \,  and  a  minimum  when  x=-\. 

5.  Determine  when  h  -^^  c{x  —  aY  is  a  maximum  or  mini- 
mum. 

By  (3)  of  Art.  146  we  may  remove  b,  by  (1)  c,  and  by  (4)  3; 
hence  we  have/(a:)  =  {x  ~  ay-,  .*.  /  {x)  =  2{x—a),  and  x—a=0, 
or  «j  =  a;  and  since /'(—  oo )  is  — ^  the  given  function  is  a  min- 
imum when  X  =  e. 

(x4-3y 

6.  Find  the  maximum  and  minimum  values  otf{x)  =  ^- 


(^+2)'- 


Hero  /'W  =  f^f 


I.  f'{x)  =  0  gives  x{x  +  3)^  =  0,  of  which  one  root  is  0  and 
the  other  two  are  —  3  and  —  3. 

II.  f^{x)  =  CO  gives  {x  -{-2y  =  0,  the  three  roots  of  which  is 
each  —  2. 

Rejecting  the  tico  equal  roots,  we  have  a^  =  —  2,  %^  =  0;  and 
since  /'(—  co)  is  +?  ./(—  2)  =  co  is  a  maximum  value  of  f{x), 
and/(0)  =  ^-  is  a  minimum. 

7.  If  the  derivative  of  f{x)  is  f'{x)  =  x^  —  10a;  +  21,  what 
values  of  x  will  render  /(a;)  a  maximum  or  minimum. 

The  roots  of  x^  —  lOx  -|-  21  =  0  are  3  and  7;  substituting 
these  for  x  mf'{x)  =  2x  —  10,  we  have/"(3)  =  G  -  10  =  -  4, 
and/"(7)  =  14  —  10  =  +  4;  therefore /(3)  is  a  maximum  and 
f{7)  is  a  minimum. 

Find  the  values  of  x  which  will  give  maximum  and  mini- 
mum values  of  the  following  functions: 


MAXIMA  AND  MINIMA.  113 

8.  u  =  x'  —  82;  +  12.  x  =  4,  min. 

9.  u  =  x^  —  3a;'  —  24:r  +  85.  x  =  —  %,  max.    a;  =  4,  min. 

10.  u  =  a;'  —  3a;'  +  6a;  +  7  Neither  a  max  nor  min. 

11.  u  =  2x'-  21a;'+  36a;  -  20.  x  =  1,  max..  a;  =  6,  min. 

12.  u  =  {x  -  9y(x  -  8)\  X  =  8,  max.^  x  =  84,  min. 

^^-  '^^  ~  — X  ~  10 —  X  =  4,  max.;  x  —  16,  min. 

(x  -\-  2Y 

^^'  "  ~  (x  —  3)"'  a^  =  3,  mas  ,  x  =  13,  min. 

a;'  +  3 

15.  w  =  ^  _  ■^^-.  a;  =  —  1,  max.;  x  =  3,  min. 

,  P  1  —  a;  +  a;' 

16.  ?^  =  - — ■ — -.  r  —  4 


X  =  i^  mm. 


17    tc  = 


a 


I  -\-  X  —  x' 
{a  -  xy 

7^2^-  ^=j,min. 

9  4 

18.  ^^  =  ~,  +  3  _  ^-  X  =  9,  max.;  a;  =  1|,  min. 

19    u  =  sin  X  +  cos  a;.  a;  =  -,  max. 

4 

20.  ?^  =  sin  a;  (1  +  cos  x).  x  =  ^,  max. 

^-        __      sin  X  7t 

^^'  ""  -  1  +  tan  X  ^  =  4"'  '''^''- 

22.  ?^  =  1  _^  ^  tan  a;'  ^  =  ^°^  ''^'  °^^^- 

23.  w  =  (1  +  a;^)(7  -  a-)'.     x=\,  max.;  .^=0  and  a;=:7,  min. 

24.  If  a;  +  ?/  =  n,  what  is  the  greatest  possible  value  of  xy  ? 

Make  ii  =  xy  —  x{n  —  x)  =  nx  —  x\ 

25.  If  _y  =  mx  +  c,  find  the  least  possible  value  of  Vx''  +  ?/' 


^to'  +  1  * 
Make        w  =  Vx'  -]-  y'  =  V(l  +  m')x^  +  2?wc:r  +  c'. 


114         DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 


26.  A  merchant  bought  a  bolt  of  linen,  paying  as  many 
cents  for  each  yard  as  there  were  yards  in  the  bolt,  and  sold  it 
at  20  cts.  per  yard;  required  the  greatest  possible  profit.     11.00. 

27.  A  club  of  X  members  has  'X"  —  12a:;"  -|-  45a;  +  15  dollars 
in  its  treasury;  how  niucli  is  that  apiece  if  the  amount  is  (1) 
a  minimum?     (2)  A  maximum  ?  (1)  113.00;  (2)  123.00. 

28.  Find  the  value  of  <p  when  sin  0  —  cos  0  is  a  maximum. 

0r=  cos-i  -1^/2  =  135°. 

29.  Find  the  fraction  that  exceeds  its  square  by  the  greatest 
possible  quantity.  i. 

30.  Find  the  fraction  that  exceeds  its  nth.  power  by  the 
greatest  possible  quantity.  f\  N^^ 

31.  Find  a  number  x  such  that  its  xih  root  shall  be  a  max- 
imum. X  =  e  =  2.7182  -)-. 

32.  Find  the  altitude  of  the  maximum  rectangle  inscribed  in 
a  given  triangle. 

Let  ABC  be  the  triangle  and  HKFE  the  required  maxi- 
mum rectangle;  let  J  =^1  B,  h  =  CD, 
x=DG,  and  u  =  area  of  HKFE',  then 

u  =  EFX  DG=  {EF)x. 

To  express  EF  in  terms  of  x,  we 
have 

CD  ■  CG  .:  AB  :  EF, 

or  h  ■  h  —  X  ::  b  :  EF; 


Fig.  30. 


o  0 

.■    EF  —  -All  —  x\,  and  u  =  ^(Aa;  —  x'\,  whose  maximum  value 
h^  '  li 

is  required. 

Dropping  y,  we  have  f{x)  —  lix  —  x" ,    -.  f'{x)  =  h  —  2x  =  0, 

whence  x  =  --;  that  is,  the  altitude  of  the  maximum  rectangle 
is  one  half  of  the  altitude  of  the  triangle. 


MAXIMA  AND  MINIMA. 


115 


D 

Fig.  21. 


33.  Find  the  altitude  of  the  maximum  rectangle  inscribed 
in  a  given  parabola. 

Let  BA  C  be  the  parabola,  and  IH 
the  rectangle;  let  h  =  AD,  x  =  AE, 
y  =  EH,  and  u  =  area  of  IH. 

Then  y''  =  iax, 

and     w  =  2{ED){EH)  =  2(h  -  x)y 
=  2{h  —  x)Viax^ 
=  4:Va  Qix^  -  x^)' 
Hence    f{x)  =  hx^  —  x^,    and    f'{x)  =  ^hx~^  —  fa;*  =  0, 

or  -—r=  SVx;  whence  x  =  ^h;    '.  DE  =  f/i. 

Vx 

34.  Find  the  altitude  of  a  maximum  cylinder  with  respect  to 
its  volume  that  can  be  inscribed  in  a  given  right  cone. 

Let  ED  be  the  altitude  of  the  cylinder 
inscribed  in  the  cone  DAC.  Let  BD  =  h, 
AD  =  I/,  ED  =  X,  EF  =  y,  and  v  =  vol- 
ume of  the  cylinder;  then 

V  =  7i{EFyED  -  ny'^x. 

To  express  y  in  terms  of  x,  we  have 

AD  :  BD  w  AE  \  EF,   or  h:b  wh  -  x  :  y, 

whence 


?/  =  jih  —  x),    and     v  =  ttj^  {h—xyx. 

.-.  f{x)  =  {h-  xYx  =  h'x  -  2hx'  +  x% 

which  is  to  be  a  maximum. 

f\x)  =  ¥  —  4:hx  +  3.-r'  =  0;     whence     Sx  -  7i  =  0,  or  a:  =  iJi. 

That  is,  the  altitude  of    the  cylinder  is  -J  of  the  altitude  of 
the  cone. 

35.  Find  the  altitude  of  the  cylinder  in  Fig.  22,  if  the  cyl- 
inder is  a  maximum  with  respect  to  its  lateral  surface. 


116        DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

Denoting  the  lateral  surface  of  the  cylinder  by  ;S',  we  have 
S  =■  ^Tr{EF)ED  =  2,nyx  =  27r- {7i  —  x)x,  which  is  to  be  a  max- 
imum.    Dropping  the  constant  factor  2;r^,  we  have 

t  {x)  =  hx  —  x'';     .'  f'{x)  =  h  —  2x  =  0,  or  X  =  ~ 

36.  Find  the  dimensions  of  a  cylindrical  open-top  vessel 
which  has  the  least  surface  with  a  given  capacity. 

Let  X  =  the  radius  of  the  base,  y  =  the  altitude,  5  =  the 
surface,  and  c  =  the  capacity. 

Then      c  =  nx^y  ...  (1)     and     s  =  ttx''  +  27rxij.  ...  (3) 

c                               2c 
From  (1),  y  =  — ^, ;     .•.  s  =  ttx'^  -\ ,  which  is  to  be  a  min- 


imum. 


■,f(x)  =  7rx'  +  -,     f\x)=27TX-~,=0', 


X 


whence  x  =  y  —,  and  y  =  y   -  ,  which  is  obtained  by  substitut- 

7t  71 

inff  for  xva.  V  ^  — t-. 

*  ^  71X 

37.  A  rectangle  is  inscribed  in  a  circle  whose  radius  is  R; 
find  the  sides  of  the  rectangle  when  it  is  a  maximum  (1)  with 
respect  to  its  area,  (2)  with  respect  to  its  perimeter. 

Each  side  =  R  V2  in  both  cases. 

38.  The  hypothenuse  of  a  right  triangle  is  A;  find  the  ratio 
of  the  other  sides  when  the  triangle  is  a  maximum  (1)  with  re- 
spect to  its  area,  (2)  with  respect  to  its  perimeter. 

Eatio  =  1  in  both  cases. 

39.  A  cylinder  is  inscribed  in  a  sphere  whose  radius  is  R', 
find  the  radius  of  the  cylinder  when  it  is  a  maximum  (1)  with 
respect  to  its  volume,  (2)  with  res|)ect  to  its  convex  surface. 

(1)  \VqR;  (2)  ^V2R. 


MAXIMA  AND  MINIMA.  117 

40.  A  cone  is  inscribed  in  a  sphere  whose  radius  is  R:,  find 
the  altitude  of  the  cone  when  it  is  a  maximum  (1)  with  respect 
to  its  volume,  (2)  with  respect  to  its  convex  surface. 

:i)  -^R;  (3)  ^R. 

41.  Find  the  maximum  isosceles  triangle  with  respect  to  its 
area  that  can  be  inscribed  in  a  given  circle. 

An  equilateral  triangle. 

42.  Find  the  dimensions  of  a  cone  whic'v?,  has  the  greatest 
volume  with  a  given  amount  of  surface. 

The  slant  height  is  three  times  the  radius  ot  the  base. 

43    Find  the  shortest  distance  from  the  point  {z'  =  1,  y'  = 

2)  to  the  line  3^  -  4:X  +  12.  Aus.  2. 

44.  Find  the  shortest  distance  from  the  poir-"t  (x'  =  2,  ?/'  = 
1)  to  the  parabola  y^  =  ix.  4/^ 

45.  A  square  sheet  of  tin  has  a  square  cut  out  at  each 
corner,  find  the  side  of  the  square  cut  out  when  the  remainder 
of  the  sheet  will  form  an  open-top  box  of  maximum  capacity. 

A  side  =  ^  the  side  ot  the  sheet  of  tin. 

46.  A  man  is  at  one  corner  of  a  square  field  whose  sides  are 
each  780  yards  and  wishes  to  go  to  the  opposite  corner  in  the 
least  possible  time;  (1)  how  far  along  the  side  must  he  go  before 
turning  across  the  field  if  he  can  travel  along  the  side  and 
through  the  field  at  the  rates,  respectively,  of  65  and  25  yards 
per  minute  ?  (2)  In  what  time  will  he  reach  the  opposite 
corner?  (1)  455  yards;  (2)  40  min.  48  sec. 

47.  Find  the  altitude  of  the  least  isosceles  triangle  circum- 
scribed about  an  ellipse  whose  semi-axes  are  a  and  b,  the  base  of 
the  triangle  being  parallel  to  the  major  axis.  3b. 

48.  A  steamer  whose  speed  is  8  knots  per  hour  and  course 
due  north  sights  another  steamer  directly  ahead,  whose  speed  is 
10  knots  and  whose  course  is  due  west.  What  must  be  the 
course  of  the  first  steamer  to  cross  the  track  of  the  second  at 
the  least  possible  distance  from  her  ?  N.  (cos"^  |)  W. 

49  If  the  statue  of  Washington  on  the  cupola  of  the  Capitol 
is  a  feet  in  height  and  b  feet  above  the  level  of  an  observer's 
eyes,  at  what  horizontal  distance  from  the  centre  of  the  cupola 


118         DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 
should  the  observer  stand  to  obtain  the  most  favorable  view  of 


the  statue?  \/h{a  -\-  b)  feet. 

MAXIMA  AND  MINIMA  OF  FUNCTIONS  OF  TWO  INDEPENDENT 

VARIABLES. 

147  Definition. — A  function,  u  =  f{x,  y),  of  two  independ- 
ent variables  has  a  maximum  or  minimum  value  according  as 

/(a;  -^h,y-^  h)  <  f(x,  y),    or    f{x  +  Ji,  y  +  k)  >  f{x,  y), 

for  all  small  values  of  li  and  k,  positive  or  negative. 

148.  Conditions  for  maxima  and  minima. — In  the  function 
u  =  f{x,  y)  if  we  suppose  x  and  y  to  vary  simultaneously,  it  is 
obvious  from  Art.  139,  that  the  maximum  or  minimum  values 
of  it-  will  occur  at  the  points  where  the  total  differential  of  u, 
[du'],  is  equal  to  zero.     That  is,  when 

'^^^^'"^^^  +  ^^^"^^ ^^^ 

As  dx  (=  h)  and  dy  {=  k)  are  independent  of  each  other, 
each  term  of  (1)  must  be  equal  to  zero.     Hence 

^=0,    and     ^=0 (2) 

These  equations  express  the  first  conditions  essential  to  the 
existence  of  either  a  maximum  or  a  minimum. 

Again,  as  u  passes  through  a  maximum  or  minimum  value, 
[du]  changes  from  +  to  — ,  or  —  to  +,  respectively  ;  therefore, 
in  the  former  case  [dul  is  decreasing,  hence  [d'^ti]  is  — ,  and  in 
the  latter  [du]  is  increasing,  hence  [d^it]  is  +•   But  the  signs  of 

must  evidently  be  independent  of  the  signs  of  dx  and  dy,  how- 
ever large  or  small  these  differentials  may  be  supposed  to  be. 
This  can  be  the  case  only  when 

[d\i\UVu\       [   d\  V 

\dx'  l\dy'  J^Kdxdyl ^^ 


MAXIMA  AND  MINIMA.  119 

For.  makmgJ  =  ^^T,    B  = -^^.     C  =  ^  ,  we  have 
AW  +  ,BkIc  +  Ck'  =  ^^'  +  ^">'  +  <^^  -  ^')A-.    .    (5) 

In  order  that  (5)  may  preserve  the  same  sign  for  all  values  of 
h  and  k,  it  is  necessary  that  AC  —  B"  should  be  positive  ;  for  if 
negative,  the  numerator  of  (5)  will  be  positive  when  k  =  0,  and 
negative  when  AJi  -\-  Bk  =  0.  Hence  we  have  as  an  additional 
condition  for  a  maximum  or  a  minimum,  AC  >  5^.  or  (4). 

With  this  condition,  the  sign  of  (5)  depends  on  that  of  the 
denominatoi  A.     Hence  for  a  maximum  we  must  have 


and  for  a  minimum 


^°'5?>'' ('> 


It  should  be  noticed  that  AC>  B"  requires  that  A  and  0 
should  have  the  same  sign. 

The  exceptional  cases,  where  B'^  =  AG,  or  where  A  =  0, 
B  =  0,  C  =  0,  require  further  investigation,  but  we  shall  not 
consider  them  in  this  book. 

The  conditions  for  a  maximum  or  minimum  value  of 
u  =  f{x,  y)  are  then,  viz.  r 

For  either  a  maximum  or  a  minimum, 

du       -           ^     du       ^  ,„. 

^  =  0,    and     ^=0; (8) 

''^'''  -Mdf>[d^)-    ■    •   ■   ■   ■     (8) 

For  a  maximum,  -^  <  0,     j-^  <  0.      ,     .    .    ,     ,     (10) 


dx^         '     dy' 

d\i  d\ 

dx'  ^    '    dy' 


For  a  minimum,  7;:;;^  >  0,     ^-^  >  0 (11) 


120        DIFFERENTIAL  AND   INTEGRAL  CALCULUS. 

EXAMPLES. 

1.  Find  the  minimum  value  of  n  —  x^  -\-  y^  —  %axy. 

Here         i—  =  3a;'  —  3«y;     —  =  3f  —  Sax; 

d^ti        ^        dh(       ^  d^u 

also  T-^  =  6x,     -^— r-  =  6?/,     -^ — T--  =  —  6a. 

ax  ay  -       ax  ay 

Applying  (8),  we  have 

x"^  —  ay  =  0,     and     y''  —  ax  —  0 ; 

whence  a;  =  0,  y  =  0 ;     and     x  =  a,  y  =  a. 

TJie  values  x  =  0,  y  =  0,  give 

d'u  _         d'^u  _  ^'w_  _  _ 

dx^  ~    '     dy""  ~    '    dxdy  ' 

Avhich  do  not  satisfy  (9).     Hence  they  do  not  give  a  maximum 
or  a  minimum. 

The  values  x  —  a,  y  —  a,  give 

d'^u  _  dSi  _  d^u    _  _ 

dx'   ~      '     dy"^  ~"      '     dxdy  ' 

which  satisfy  both  (9)  and  (11). 

Hence  they  give  a  minimum  value  of  u,  which  is  —  a^. 

2.  Find  the  minimum  value  of 

x"  +  xy  -\r  y""  —  ax  —  by.  i{ab  —  a'  —  &'). 

3.  Find  the  maximum  value  of 

{a  -x){a-  y) {x -\- y  -  a).  g-^. 

4.  Required  the  triangle  of  maximum  area  that  can  be  inscribed 
in  a  given  circle.  The  triangle  is  equilateral. 

5.  Divide  a  into  three  parts,  x,y,  a  —  x  —  y,  such  that  their 
continued  product,  xy(^a  —  x  —  y),  may  be  the  greatest  possible. 

x  =  y  =  a-x  —  y  =  -. 


MAXIMA   AND  MINIMA  121 

6.  Divide  45  into  three  parts,  x,  y,  4,5  —  x  —  y,  such  that 
x''y^{4:5  —  X  —  yY  may  be  a  maximum. 

X  =  10,  y=  15,  '45  —  X  —  y  =  20. 

7.  Find  the  least  possible  surface  of  a  rectangular  parallele- 
piped whose  volume  is  «^  6a'". 

8.  Find  the  dimensions  of  the  greatest  rectangular  parallelo- 

x'         11^         z^ 

piped  that  can  be  inscribed  in  the  ellipsoid  ^  +  75-  +  -5  =  1. 

fa  4/3,  |&  4/3,  Ic  V3~ 


CHAPTER    VII. 

APPLICATIONS    OF    THE     DIFFERENTIAL    CALCULUS 
TO    PLANE    CURVES. 

TANGENTS,    NORMALS,   AND  ASYMPTOTES. 

149.  Equations  of  tlie  Tangent  and  Normal.  In  Fig.  23 
let  P(«,,  y,)  be  the  point  of  tangency  of  the  tangent  TP;  then 
the  equation  of   TP  is  y  —  y^  =^  m{x  —  x^,  where  m   is  the 

tangent  of  the  angle  BTP.      But,  Art.  26,  tan  BTP  =  -^\ 

therefore  the  equation  of  the  tangent  PT\^ 

^~^'^^^^~^'^'     ••••••     (^) 

where  -^  is  the  value  of  -^-  with  respect  to  the  curve  ^P  at 
dx^  ax 

the  point  {x^ ,  ?/,), 

Since  the  normal  PN  is  perpendicular  to  the  tangent  or 

curve  at  P,  its  equation  may  be  obtained  from  (A)  by  substitut- 

dx^  »      dy^  . 

mg 7—  lor  -j-^,  which  gives 

y-y.=  -  ^(^  -  «i) (^) 


EXAMPLES. 

1.  Find  the  equations  of  the  tangent  and  normal  to  the  pa- 
rabola y"^  =  iax. 

123 


DIFFERENTIAL  CALCULUS  AND  PLANE  CURVES,     123 

„  dy       2a  dy,       2a 

Here  -^  =  —  ;    .*•  -f^  =  — . 

dx        y  dx^       y, 

dy 
Substituting  this  value  of  ~  in  (A)  and  (B),  we  have 


y  -y,^  -^(^  -  X,),  tangent;    ....     (1) 


2/-.y,  = -|^(rr-a;J,  normal.      ...     (2) 


2.  Find  the  equations  of  the  tangent  and  normal  to  the  pa- 
rabola y"^  =  ISx  at  the  point  a;,  =  2. 

Here     Aa  =  18  and  y,^  =  ISa;^;     .'.  2a  =  9  and  y,  =  6. 
Substituting  in  (1)  and  (2),  and  reducing,  we  have 

'Zy  =  ?>x  -\-  6,  tangent,     and     "dy  =  —  2x  -{-  22,  normal. 

Find  the  equations  of  the  tangent  and  normal  to  the  follow- 
ing curves: 

3.  The  circle,  ?/'  +  x"  =  R\ 

(1)  yy^  +  a^^i  =  ^';    (2)  y^^  -  ^y^  =  0. 

4.  The  ellipse,  ay  +  b'x'  =  d'h\ 

(1)  a-^yy,  +  h^xx^  =  a^b';     (2)  y-y,  =  -^{x  -  x,). 

5.  The  cissoid,  y'^(2a  —  x)  —  x^.  • 

(1)  tangent,  y-y^  =  ±       ,  ~  ""'^  .f'C^-  -  ^J- 

(2a  -  a;  J' 


6.  Find  the  equation  of  the  normal  to  y'''  =  6a;  —  5  at  ^^  =  5. 

40 
3' 


^  =  -  l«  +  -*'"- 


124        DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

7.  Find  the  equation  of  the  tangent  at  ^/i  =  4  to  the  cycloid 

x=lO  vers-i  ^  —  V'^Oy—  y\ 

?/  =  2a; +20(1 -vers -If). 

150.  Length  of  Tangent,  Normal,  Subtangent,  and  Sub- 
normal.    Let  PT represent  the  tangent  at  the  point  P{x^,  y^), 
PiV^the  normal;  then  y^  =  FB,  BT  is  the 
subtangent,  and  BN  is  the  subnormal. 
Let  0  =  the  angle  BTP,  then 


T        0  A  B 

Pig.  23. 

that  is, 

(3) 

that  is, 

(3) 

that  is, 

(4) 

that  is. 


N    (1)  BT=  BP  cot  (p  =  y^ 


dx. 


subtangent  =  V-^t^- 


(C) 


BN  ~  BP  tan  BPN  =  BP  tan  0; 


subnormal  =  y, 


dx. 


ds. 


TP  ^{BP~  sin  cp)  =  BP-^; 


ds. 


.     .     (D) 

(Art.  49) 


tangent  =  y,^ (E) 


PN  ~  (BP  -^  cos  0)  =  BP 


ds^ 
dx. 


normal  =  y^ 


ds, 
dx. 


(F) 


In  formulas  (E)  and  (F),  ds^  =  Vdx^'  +  dy^\  Art.  33. 

If  the  subtangent  be  estimated  from  the  point  T,  and  the 
subnormal  from  B,  each  will  be  positive  pr  negative  according 
as  it  extends  to  the  right  or  left. 


DIFFERENTIAL  CALCULUS  AND  PLANE  CUBVES.     125 

EXAMPLES. 

1.  Find  the  values  of  the  subtangent  and  subnormal  of  the 
ellipse  ay  +  6V  =  a'b\ 


(III  Vx 

Here  -^  = j-^;  substituting  in  (C)  and  (D)  we  have 

dx^  a  y  ^ 


Subt.  =  —  -r~-  =  — ;    subn.  = 


b'x. 


^ 

2.  Find  the  values  of  the  sub  tangent  and  subnormal  of  the 
ellipse  9_?/^  +  4:X'  =  36,  at  x^  =  1. 

Here  a  =  'ii,  b  :=  2,  x^  =  1,  which  substituted  in  the  preced- 
ing answers  give  subt.  =  —  8,  subn.  =  —  |. 

Find  the  values  of  the  subtangents  and  subnormals  of  the 
following: 

3.  v'  =  «a:.  Subt.  =  3a;, ;        subn.  =  ;r— . 
^  "  3y, 

4.  Parabola,  y^  =  4^ax.       Subt.  =  2x^;        subn.  =  2a. 

5.  y  =  a".  Subt.  =  = ;      subn.  =  a^i  log  a. 

^  log  a'  ^ 


6.  Find  the  length  of  the  tangent  of  the  tractrix. 


x  =  a\og  [^-^^^^      '^)  -  (a'  -  f)'^       Tang.  =  a. 

7.  Find   the  lengths  of  the  normal  and  subnormal  of  the 
cycloid  X  =  r  vers"^  -  —  ^2ry  —  ^^ 

Norm.  =  4/(3r?/) ;  subn.  =  \/{2ry  —  ?/'). 


126         DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 


151.  Lengths  of  Tangent,  Normal,  Subtangent,  and  Sub- 
normal in  Polar  Co-ordinates.  Let 
AP  (=  s)  be  a  curve,  0  the  pole,  OP 
(=  r)  the  radius  vector,  PT  a  tangent 
at  P.     Let  e  =  XOP. 

Draw  OT  perpendicular  to  OP 
and  prolong  it  to  meet  the  normal 
"^  NP  at  N;  then  PT'  is  the  polar  tan- 
gent, PN  the  polar  normal,  OT  the 
polar  subtangent,  and  ON  the  polar 
subnormal.     Evidently, 


Fig.  24. 


(1) 

that  is, 

(2) 

that  is, 

(3) 

that  is, 

(4) 

that  is, 


0T=  OPi&.uOPT=7 


ONP  =  OPT=  tp.        (Art.  97) 
rcU 


dr  r 


the  polar  subtangent  =  -y^. 


(Art.  98) 
.    .     (G) 


ON  =  OP  cot  ONP  =  r 


dr 

Vdor 


dr 


the  polar  subnormal  =  -jj: 


(H) 


dv 
TP  =  {OP  -^  cos  OPT)  =  r^j^', 

the  polar  tangent  =  —j^ (I) 


dr 


PN  ={OP-T-  sin  ONP)  =r~ 


ds 


vdd  ^ 
ds  ' 


the  polar  normal  =  yj 


(J) 


In  formulas  (I)  and  (J)  ds  =  Vdr'  +  r'dO',  Art.  97. 


DIFFERENTIAL   CALCULUS  AND  PLANE  CURVES.     127 

EXAMPLES. 

Find  the  tangent,  normal,  subtangeut,  and  subnormal  of  the 
following  polar  curves: 

1.  The  spiral  of  Archimedes,  r  =  ad. 

dr      a ' 


r 
From  (G),  subt,    =  —  ;    from  (H),  subn,  =  a\ 

CI 


from  (I),    tang.  =  —  |/a°  -f  f'^l 


from  (J),  norm.  =  Va^  -\-  r^. 
1    The  logarithmic  spiral,  r  =  a^. 

^  =  anoga. 

Substituting  in  (G),  (H),  (I),  (J),  we  find 


subt.  =  q =  mr;     subn.  =  — ; 

loff  a  m 


tang.  =  r  Vl-{-  m^ ;    norm.  =  r  Vl  -\-  log''  a. 

Find  the  subtangent  and  subnormal  of  the  following: 

3.  The  hyperbolic  spiral,  7'd  —  a. 

Subt.  —  —  « ;  subn.  = . 

a 

4.  The  Lemniscate  of  Bernouilli,  r"^  =  a^  cos  26^. 

Subt.  =  —^--. — — n;    subn.  = sin  2^. 

a"  sm  26^'  r 


128        DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

152.  An  Asymptote  to  a  curve  is  a  tangent  which  passes 
within  a  finite  distance  of  the  origin  and  touches  the  curve  at  an 
infinite  distance.  A  curve  which  has  no  infinite  branch  can 
have  no  real  asymptote. 

In  Fig.  23,  let  x^  and  y^  represent  the  intercepts  OTand  OD, 
respectively;  then,  in  (A),  Art.  149,  by  making  [\)  y  =  0  and 
(2)  a;  =  0,  we  find 

{l)x,  =  x,-y!^=OT;      .    .    .    .     (K) 
{^)y.  =  y^-^!^'  =  OD (L) 


Now,  if  the  curve  AP  is  of  such  a  character  that  x^  or  y^ ,  or 
both,  remain  finite  wlien  x^  or  y^,  or  both,  become  infinite  (see 
Art.  154),  the  tangent  TF  will  be  an  asymptote  to  the  curve. 

EXAMPLES. 
1.  Examine  y^  =  6«^  +  x^  for  asymptotes.  ^ 

cli/       ^x    I    x^   civ         ^X     \    X 
Since  -#  =  — A^ — ,  -^  =     '  '  ,  which  substituted  in 


dx  y'      '  dx^  y^ 

(K)  and  (L)  give 

(1)  x,=x^-   y^ 


2  3 


4a;,  ^x^  £       ^ 

x^ 

which  =  —  2  when  x^  =  co . 

i^\        -  4V  +  a;,^  _  2 

which  =  2  when  a;,  =  c» . 


DIFFERENTIAL   CALCULUS  AND  PLANE  CURVES.     129 

Therefore  the  straight  line  whose  x  and  y  intercepts  are  —  2 
and  +  3j  respectively,  is  an  asymptote  to  the  curve. 

Since  the  asymptote  passes  through  the  points  (—  2,  0)  and 
(0,  2),  its  equation  is  ?/  =  x  -\-  2. 


153.  General  Equation  of  the  Asymptote.  Since  the 
asymptote  passes  through  the  points  {x^ ,  0)  and  (0,  y J,  its  equa- 
tion is 


This  equation  enables  us  to  determine  whether  or  not  any 

given  curve  has  an  asymptote,  and,  if  it  has,  to  find  its  equation. 

dy 
Let  us  denote  the  values  which  -y--  and  y^  assume  when  a;j  =  oo 

by  wij  and  1)^ ,  respectively;  then  we  have 

y  =  m^x  +  5, (P) 

154.  When  the  terms  of  the  equation  f{x,  y)  =  0  are  of 
different  degrees,  to  find  the  relation  of  y  to  x  when  they  are 
infinite,  we  may  omit  all  the  terms  except  the  group  which  are 
of  the  highest  degree  with  respect  to  x  and  y. 

Thus,  when  x  is  infinite,  the  equation  ay''  —  hx^  -\-  cy  -\-  dx  =  e 


=  ^A 


gives  ay''  —  Ix'  =  0,  or  y  =  ±  y  —x. 

A  curve  like  a-y''  +  ^''^^  =  «'^%  or  y*  =  x''{a^  —  x^),  etc., 
which  has  no  infinite  branch  or  branches,  has  no  real  asymptote; 
this  is  indicated  by  the  fact  that  when  x  is  infinite,  y,  as  deter- 
mined above,  will  be  imaginary. 

2.  Find  the  asymptote  of  the  hyperbola  a'^y''  —  Vx'  =  —  a^b''. 

h  b 

When  X  =  cc ,    y  =  ±  -x,    or    ^/^  =  ±  -x^. 


130        DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 


.-.  m^  =  ±  -  and  5j  =  0,  which  substituted  in  (P)  gives 

3.  Find  the  asymptote  of  the  parabola  y"  =  iax. 

When  X  =  cc ,     y  =  ±  2  Vax,     or     y^=  ±  2  Vax^. 

dy^        2a  2ax^         , —        ,  .  ,  , 

-~  =  —  ;     w„  =  w, =  \  ax, ,  which  =  co  when  a;,  =  00 . 

Therefore  the  parabola  has  no  asymptote. 

4.  Find  the  asymptote  of  y'^  —  ax^  +  a;'. 

When  a;  =  00 ,  we  have  y^  =  x"^;  .:  y  =  x  or  y^  =  x^. 

hence  m^  =  1  and  5j  =     ,  and  the  asymptote  is  ?/  =  .r  +  — . 
o  o 

155.  Asymptotes  Determined  by  Inspection.     When  an 

asymptote  is  perpendicular  to  the  axis  of  x  or  y,  it  can  often  be 

determined  by  inspection.    In  the  first  case  m^  =  cc ,  ov  y-'  —  0, 

which,  substituted  in  (M),  gives  x  —  x^  =  0,  since,  in  this  case, 
Xg=x^;  that  is,  if  y^  is  infinite  when  a;^  is  finite,  a;  —  «,  =  0  is 
the  equation  of  the  asymptote. 

x" 

Thus,  in  the  cissoid,     ?/°  = , 

•^        2a  —  X 

y  =  <x)  when  x  —  2a;  hence  the  line  x  —  Za  —.  0,  which  is  paral- 


DIFFERENTIAL   CALCULUS  AND  PLANE  CURVES.     131 

lei  to  the  axis  of  y  and  at  a  distance  2«  from  it,  is  an  asymptote 
to  the  curve. 

Again,  in  xy  =  a  or  y  =^  —,  when  a;  =  0,  y  =  (x, ;    therefore 

X  =  0,  or  the  axis  of  y,  is  an  asymptote  to  the  curve. 

Similarly,  in  y  =  a^,  when  x  =  —  co ,  ?/  =  0;  hence  y  =  0, 
or  the  axis  of  x,  is  an  asymptote  to  the  logarithmic  curve. 

5.  Find  the  asymptotes  of  xy  —  ay  —  hx  —  0. 

(1)  X  -  a  =  0;     {2)  y  —  b  =  0. 

6.  Find  the  asymptote  of  y^  =  ax"^  —  x\ 

7.  Find  the  asymptotes  of  y  =  c  -\-  j-^ 


{x  -  6)^' 

y  =■  c  and  x  ^=  b. 

8.  Find  the  asymptotes  of  ?/^(a;^  -f  1)  =r  a:'(«^  —  1). 

y  =  ±x. 

9.  Find  the  asymptotes  of  y''{x  —  a)  =  x^  -{-  ax\ 

X  =  a  and  y  =  ±  [x  -{-  a). 


CURVATURE. 

156.  A  point  moving  along  an  arc  of  a  curve  changes  its 
direction  continuously,  and  the  total  change  of  direction  is  called 
the  Total  Curvature  of  the  arc. 


The  angle  TtP',  Fig.  25,  through  which  the  tangent  PT 
rotates  as  the  point  of  tangency  P  moves  from  P  to  P',  being 


132        DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

the  total  change  of  direction  of  the  point  P,  is  the  total  curva- 
ture of  the  arc  PP'. 

157.  ITniform  Curvature.  The  curvature  is  uniform  when, 
as  the  point  of  tangency  moves  over  equal  arcs,  the  tangent 
turns  through  equal  angles;  that  is,  when  the  distance  described 
by  the  point  varies  as  its  direction. 

Let  APm  be  the  curve,  AP  =  s,  PP'  =  As,  XEP  =  0, 
Art.  49;  then  TtP'  =  J0.  Let  PU  and  P'C'be  normals  meet- 
ing at  C. 

Supposing  As  oc  A(p,  we  have  (Art.  12) 

As  =  mAd),     or     —  =  -^. 
^  m       As 

(1)  Let  us  consider  the  meaning  of  —t-.  If  the  distance  As 
gives  a  total  curvature  of  Acp,  since  As  cc  A(p,  a  distance  of  1 

will  give  a  curvature  of  --r-*     That  is,  --r~    is   the   curvature 

^  As  As 

per  distance  of  unity,  or  the  rate  of  change  of  the  direction  of  a 
curve  with  respect  to  that  of  its  length,  for  which  reason  it  is 
called  the  curvature  of  the  curve, 

(2)  Let  us  determine  the  value  of  m.  The  circle  is  the  only 
curve  of  uniform  curvature.  Hence,  supposing  As  a  Acp,  PP' 
is  the  arc  of  a  circle  whose  radius  (say  E)  is  CP.  The  angle 
POP'  =  TtP'  =  Acp;  but  arc  PP'  =  CP  X  angle  PCP';  that 
is.  As  —  RA(p\  hence  m  =  R,  and  we  have 

Acp  _  1^ 
As    '^  R' 

Cor.  L  The  curvature  of  any  circle  is  equal  to  the  reciprocal 
of  its  radius;  and  the  curvatures  of  any  two  circles  are  inversely 
proportional  to  their  radii. 

Cor.  II.  If  -ff  =  1,  -~—  1;  that  is,  the  unit  of  curvature 

is  the  curvature  of  a  circle  whose  radius  is  unity. 


DIFFERENTIAL   CALCULUS  AND  PLANE  CURVES.     133 

158.  Variable  Curvature.  When  the  curvature  is  variable, 
Ave  define  the  curvature  at  any  point  F  of  the  curve  as  the  value 

which  — —  would  have  were  the  curvature  there  to  become  uni- 

form.   Hence  the  curvature  at  P  is  the  value  of  -7-  at  that  point. 

as 

159.  Radius  of  Curvature.  A  circle  tangent  to  a  curve  at 
any  point,  and  having  the  same  curvature  as  that  of  the  curve 
at  that  point,  is  called  the  circle  of  curvature;  its  radius,  the 
radius  of  curvature;  and  its  centre,  the  centre  of  curvature. 

The  curvature  of  this  circle  being  that  of  the  given  curve,  is 

equal  to  -j-;  therefore  the  radius  of  curvature  of  A  Pin  at  P, 
Fig.  25,  is 


Cor.  I.  To  express  R  in  terms  of  the  differentials  of  x  and  y. 

tan  d)  =  ^,         .•.  0  =  tan-^  -f^; 
dx  dx' 


hence  we  have 


d^v  dx  1 

<14>  =  ^^,  ,  ^  .  ;    also,   ds  =  {dx"  +  diff. 


1  +  f^)? 
,.^.  =  ^  =  i^_^,    or    \Z}^.     .     (1) 
d(p  dxdy  d^y 

d^ 


*  R  will  be  positive  or  negative  according  as  the  curve  is  convex  or 
concave  (Art.  173),  but  its  sign  is  often  neglected. 


134        DIFFERENTIAL  AND  INTEGBAL  CALCULUS. 

EXAMPLES. 

1.  Find  the  radius  of  curvature  of  the  parabola  y^  =  ^ax. 

Here  -t~  —  —■>    and    -y^  = j-. 

ax       y  ax'  y 

Substituting  in  (1),  we  have 

^ la^ • 

At  the  vertex,  where  y  =  0,  we  have  B  =  2a,  which  is  evi- 
dently the  minimum  radius  of  curvature, 

2.  Find  the  radius  of  curvature  of  the  ellipse  a^y^  -\-  JV  = 
aW. 

dy  _       Vx      d'y  _         h* 
dx  ~       a^y'    7lx'  ~      a'y^ 


,2^.3    J 


i)'  [       ay      J  ~  a'b' 

At  the  vertex  x  =^  a,  y  =  0,  R  =  — ,  and  at  the  vertex  a;  =  0, 

a" 
y  :=  h,  R  =  Y>  which  are  respectively  the  minimum  and  maxi- 
mum radii  of  curvature. 

3.  Find  the  radius  of  curvature  of  the  cycloid 

/it  . 

X  =^  r  vers~^  -  —  V2ry  —  y^. 


r 


Here         v-  = 


(ly       V2ry-y''    "      ~^'^«'       y' 


DIFFERENTIAL  CALCULUS  AND  PLANE  CURVES.     135 

^  —  —  —  •       •    R  —  2  V^riT 

which  equals  twice  the  normal. 

4.  Find  the  radius  of  curvature  of  the  logarithmic  curve 


7711J 

5.  Find  the  point  on  the  parabola  i/^  =  8x  at  which  the 
radius  of  curvature  is  7||.  y  =  S,  x  =  1^. 

6.  Find  the  radius  of  curvature  ot  y  =  x*  —  ix^  —  ISx"^  at 
the  origin.  1 

^  ^  36  • 

7.  Find  the  curvature  of  the  equilateral  hyperbola  xy  =  12 
at  the  point  where  x  ^=  3.  1        24 

E  ^125' 

8.  Find  the  radius  of  curvature  of  the  catenary 

n  I  ^  _  *\ 


y 


9.  Find  the  radius  of  curvature  of  the  hypocycloid 

x^  ^y"^  =  a^.  R  =  3{axyf 


160.  The  radius  of  curvature  in  polar  co-ordinates  can 

be  found  by  transforming  the  value  of  R  given  in  the  answer  to 
Ex.  7,  Art.  112,  to  polar  co-ordinates.     We  thus  obtain 

.    clr-" 


^  =  777*^ — ^ ^'^ 

r  -\-  2 r • 


136        DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 


EXAMPLES. 

1.  Find  the  radius  of  curvature  of  the  spiral  of  Archimedes, 

r  =  ad. 

TT  dr  a}r       . 


substituting  in  (2),  we  have 


R 


{r'  +  ci^y^  _  a{\  +  ff'f 


Find  the  radius  of  curvature  of  the  following: 

2.  The  logarithmic  spiral  r  =  a^.  ^  =  r  4/I  +  (log  ay 


o 


3.  The  cardioid  r  =  a{l  —  cos  6).  ^  —  ^  \^2ar. 


4.  The  lemniscate  r'  ==  a'  cos  20.  E  = 


3r 


CONTACT  OF   DIFFERENT   ORDERS. 

161.  Let  y  =  f[x)  and  y  =  ^{x)  be  any  two  curves  referred 
to  the  same  axes.  Let  the  curves  intersect  at  the  point  P, 
whose  abscissa  is  a,  then  f{a)  =  (p{a).  If  /(a)  =  <p{a),  and 
f'(a)  =  (p'{a),  the  curves  are  tangent  at  F,  and  are  said  to  have 
a  contact  of  the  first  order.  If/(a)  =  (p{a),f'{a)  =  <p'{a),  and 
f"{a)  =  (f)"{a),  the  curves  have  the  same  curvature  at  P,  and 
their  contact  is  of  the  second  order.  If,  in  addition,  /"'(^?)  = 
0"'(a),  their  contact  is  of  the  third  order;  and  so  on.  Thus, 
contact  of  the  wth  order  imposes  n  -\- 1  conditions. 

162.  Two  curves  cross  or  do  not  cross  at  their  point  of  co7i- 
tact,  according  as  their  order  of  contact  As  even  or  odd. 

Let  a;  =  a  be  the  abscissa  of  the  point  of  contact  of  the 


DIFFERENTIAL  CALCULUS  AND  PLANE  CURVES.     137 

curres  y  =f{x)  and  y  =  <p{x),  then  f{a)  =  <p{a).    Let  h  be  a 
small  increment  of  a;.     By  Taylor's  formula,  we  have 

f{a  +  h)  =f{a)  +f'{a)n  +f"{a)^  +/'-(«)|^  +;      (i) 


0(«  +  h)  =  <p{a)  +  <p'{a)h  +  0''(a)-^  +  0"''(«)-^  +.      (3) 


Subtracting  (2)  from  (1),  we  obtain 


fia  +  ?i)-<p{a  +  h)  =  A[/'(a)-0'(a)]  +  -^[/"(a)  -  0"(a)] 


+  |^[/'"(«)  -  ^'"i")]  +  ^ir^i^)  -  0"(^O]  +.      (3) 


(a)  ltf{x)  —  (p{x)  changes  sign  as  x  increases  from  a  —  h  to 
a -{- h,  the  two  curves  evidently  cross  at  a;  if  not,  the  curves 
touch  each  other,  but  do  not  cross. 

(b)  If  the  contact  is  of  an  odd  order,  the  first  term  of  the 
second  member  of  (3),  which  does  not  vanish,  contains  an  even 
power  of  h ;  hence  the  sign  of  the  second  member,  and  therefore 
the  first,  undergoes  no  change  as  x  passes  from  a  —  h  to  a  -\-  h, 
and  the  curves  do  not  cross. 

(c)  If  the  contact  is  of  an  even  order,  the  first  term  of  the 
second  member  of  (3),  which  does  not  vanish,  contains  an  odd 
power  of  h;  hence,  in  this  case,  /"(a;)  —  (p{x)  changes  sign  as  x, 
passes  from  a  —  h  to  a  -\-  h,  and  therefore  the  curves  cross. 

CoE.  I.  At  a  point  of  maximum  or  minimum  curvature,  the 
circle  of  curvature  has  contact  of  the  third  order  with  the  curve, 
for  it  does  not  cut  the  curve  at  such  a  point. 

Cor.  II.  If  two  curves  are  tangent  to,  and  cross  each  other  at, 
a  certain  point,  they  have  contact  of  at  least  the  second  order. 


138        DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

EXAMPLES. 

1.  Find  the  order  of  contact  of  the  two  curves 

y  =  x'-  ^x'  +  7     and    y  +  ^x  =  Q. 

By  combining  the  two  equations  we  find  that  (a;  =  1,  ?/  =  5) 
is  a  point  of  contact. 

Making /(a;)  =  x^  —  ox^-\-l  and  (j){x)  —S  —  2>x,  we  have 

f'{x)  =  ^x'-%x,     0»  =  -3;     .-.     /(1)=     0'(l)=-3; 
f"{x)  =  Qx-Q,      cp"{x)=0',  .•./"(!)=    0"(1)  =  O; 

f'"{x)  =  6,  0'"(^)  -  0;  .-.  /"'(I)  >   0'"(1)- 

Hence  the  contact  is  of  the  second  order. 

2.  Find  the  order  of  contact  of  the  parabola  ?/"  =  4a;  and  the 
line  dy  =  x  +  9.  First  order. 

3.  Find  the  order  of  contact  of  the  curves 

y  =  3x  —  x"    and     xy  =  '6x  —  1.      Second  order. 

4.  Find  the  order  of  contact  of 

y  =  log  {x  -  1)     and    x""  -  6x +  2y  +  8  =  0, 

at  the  point  (2,  0).  Second  order. 

5.  Find  the  order  of  contact  of  the  parabola  y"^  =  4x  -\~  4  and 
the  circle  y'  -J^  x' =  2x  +  3.  Third  order. 


163.  Osculating  Curves.  The  curve  of  a  given  species 
that  has  the  highest  order  of  contact  possible  with  a  given  curve 
at  any  point  is  called  the  osculating  curve  of  that  species. 

A  curve  may  be  made  to  fulfil  as  many  independent  condi- 
tions as  there  are  arbitrary  constants  in  its  equation,  and  no 
more.     Therefore,  in  order  that  y=f{x)  may  have  contact  of 


DIFFEBENTIAL  CALCULUS  AND  PLANE  CURVES.    139 

the  nih.  order  with  a  given  curve  at  a  given  point,  the  equation 
must  involve  n  -\-l  arbitrary  constants. 

Hence,  as  y  =1  ax  -\-  b  has  two  constants,  the  osculating 
straight  line  has  contact  of  the  first  order. 

As  {x  —  ay  +  (y  —  by  ~  r^  has  three  constants,  the  osculat- 
ing circle  has,  in  general,  contact  of  the  second  order. 

164.  To  find  the  osculating  straight  line  at  any  point 
{x',  y')  of  a  given  curve  y  =  f{x). 

The  equation  of  a  line  is 

yz=ax  +  b (1) 

Since  the  line  and  curve  pass  through  {x',  y'),  we  have 

y'  =  ax'  +  b=f{x').    ......     (2) 

Also,  gl  =  a  =/'(A    ......    (3) 

since /'(a;')  =  <f)'{x'). 

Solving  (2)  and  (3)  for  a  and  b,  we  have 

a  =  ^,     and     b  =  y'~^^x', 

which,  substituted  in  (1),  gives 

Therefore  the  osculating  straight  line  is  a  tangent  to  the 
curve,  as  would  be  inferred. 

165.  To  find  the  radius  of  the  osculating  circle  at  any 
point  of  a  given  curve,  y  —f{x). 

The  general  equation  of  a  circle  whose  radius  is  r  is 

(x-ay-^{y-iy^r^ (1) 


140        DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

Differentiating  twice  successively,  we  have 

x-a  +  (y-li^=0, (S) 


l  +  i  +  (2'-*)S  =  » •     (5) 


From  (3),         y-i=-'!^Jl (4) 

FroM^),       -«  =  ^|31.. (^) 

Substituting  (4)  and  (5)  in  (1),  we  have 

dx  d'^y 

By  comparing  this  result  with  formula  1,  Art.  159,  it  will  he 
seen  that  the  osculating  circle  is  the  same  as  the  circle  of  cur- 
vature. 


INVOLUTES  AND   EVOLUTES. 

166.  An  Involute  may  be  regarded  as  a  curve  traced  by  a 
point  in  a  thread  as  it  is  unwound  from  another  curve,  called 
the  Evolute. 

Thus,  imagine  a  thread  stretched  around  the  curve  A^P^m^ 
with  one  end  fastened  at  m/,  if  the  thread  is  unwound  by  carry- 
ing the  point  at  A  above  and  around  to  the  right,  that  point  of 
the  thread  will  trace  the  involute  APm  of  which  A^P^m^  is  the 
evolute. 

An  evolute  may  have  an  unlimited  njimber  of  involutes,  for 
A  may  be  any  point  on  the  curve  A^m^. 


DIFFERENTIAL  CALCULUS  AND  PLANE  CURVES.    141 

lu  wliat  follows  the  chief  object  is  to  deduce  certain  prop- 
erties of  the  evolute  from  its  involute,  or  vice  ve?-sa,  and  for 
uniformity  the  co-ordinates  of  P  (the  involute)  will  be  repre- 
sented by  X,  y,  and  those  of  P,  (the  corresponding  point  of  the 


evolute)  by  ^, ,  y,;  the  arc  AP  by  s;  the  arc  AP^  by  5, ;  and  the 
angles  of  direction  of  AP  and  AP^ ,  at  P  and  P^,hj  (p  and  cp^ , 
respectively. 

167.  Elementary  Principles.     I.  PP^  =  the  arc  ^P^  =  s^. 


II.  PP,  is  tangent  to  AP^ni^  at  Pj,for  it  has  the   same 
direction  as  the  curve  at  that  point. 

III.  The  line  PP,  is  a  normal  to  the  curve  APm  at  P. 


142         DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

For,  draw  TP  tangent  to  the  curve  AP  at  P. 
{P,Ef-^{EPy  =  {P,P)\     or     {y^-yy  +  [x-x,y  =  s,\    (1) 

iVi -y){^yi- (^y) -i'{^-x,W^-(^x,)  =  s,ds^.    .   (3) 

Again,  P^U  =  P,P  sin  PPP, ,     or     y^-y  =  sf^^;    .     .     (3) 

also,        EP  =  P,P  cos  EPP, ,    or    x~x,  =  -  s"p.      .     (4) 

Substituting  in  (2)  from  (3)  and  (4),  and  reducing,  remem- 

dy  d(jc 

bering  that  dx^^  +  ^^/  =  ds^'',  we  have  —-'  =  —  — ;  that  is, 

tan  0,  ==  -  cot  0  ;    .-.  0^  =  f  +  0,   or  PFX  =  f  +  PTX; 
hence  PJ^  is  perpendicular  to  the  tangent  PT. 

Cor.  I.  Since  sin  0^  =  cos  0,        -^  =  — ;    .     .     .     .     (5) 

dx  dv 

also,  since  cos  0^  =  —  sin  0,     ^'  =  —  ~^.     .     ,  .  .     (6) 

Cor.  II.  The  point  P^  is  the  centre  of  curvature  of  the  curve 
APm  Sit  P. 

For,  if  circles  be  described  from  P/  and  P^  as  centres  with 
P^P'  and  P,P  as  radii,  respectively,  the  arc  PP'  will  lie  within 
the  one  circle  and  without  the  .other,  since  the  straight  line 
P^'P'  is  equal  to  the  partly  curved  line  P^'P^P.  Hence  the 
circumference  of  the  circle  whose  centre  is  P^  crosses  and  touches 
the  curve  APm  at  P  (Art.  162,  Cor.  II). 

Cor.  III.  Since  P^P  =  s^  =  R,  we  have  (Art.  159) 

S    ='I^=   (^^^'    +  'lf)\ (7) 

'       d(p  dxd^y       ^  ^ 


DIFFERENTIAL  CALCULUS  AND  PLANE  CURVES.    143 
Cor.  IV.  From  (4)  and  (6),    x^=x  -  sff,     ....     (8) 

and  from  (3)  and  (5),  y,  =  y^  «:^ (9) 

Substituting  for  s^  in  (8)  and  (9),  from  (7),  we  have 

^.=^ ^5 '     ^^^    ^.-2/  +  -^y-.  .(10) 

These  values  of  x^  and  y^  are  the  values  of  the  co-ordinates 
of  the  centre  of  curvature  at  the  point  P. 

168.  To  find  the  equation  of  the  e volute  of  any  given 
curve. 

By  differentiating  the  equation  of  the  given  curve,  and  sub- 
stituting the  results  in  (10),  x^  and  ?/,  may  be  expressed  in  terms 
of  X  and  y.  If,  between  the  equations  thus  obtained  and  that 
of  the  given  curve,  x  and  y  be  eliminated,  the  resulting  equation 
involving  x^  and  y^  will  be  the  equation  of  the  evolute. 


EXAMPLES. 
1.  Find  the  equation  of  the  evolute  of  the  parabola?/^  =  ^ax. 

dy  _  2a  _     d^  _       ia^ 
dx        y  '     dx''  ~       y^' 

Substituting  in  (10),  we  have 

,    «'  +  4a'    2a    ?/        -     ,  _ 

x,  =  x+  ^-^5 f-^=  3a;-f  3a; 

y  y    4:a^ 

_x'  —  2a^ 


144        DIFFEBENTIAL  AND  INTEGRAL  CALCULUS. 


ij'  +  4a^ 

f  _ 
4a= 

Ad- 

y=-  {2afy;\ 

These  values  of  x  and  y  substituted  in  ?/"  =  4:ax  give 

which  is  the  equation  required;  hence  the  evohite  is  the  semi- 
cubical  parabola. 

Cor.  I.  The  length  of  the  arc  of  the  evolute  s^  may  be  found 
by  formula  (7),  Art.  167. 

2.  Find  the  evolute  of  the  ellipse  ay  +  ^'«'  =  a'b\ 

{ax^f  +  {bi/^  =  {a'  -  h-'f. 

3.  Find  the  co-ordinates  of  the  centre  of  curvature  of  the 
cubical  parabola  y^  =  ct^x. 

J  a'  +  16  y'        _  a'y  -  9//^ 
^'  ~       Qd'y    '  ^'  ~     '  2d'   ' 

4.  Find  the  co-ordinates  of  the  centre  of  curvature  of  the 


a  I   a    .  a\  ^     =x  -'^-  W  -  fr       n,     -   9. 


catenary  y  = -[e  -\- &       .  x^=x  - ''-^  y' -  a',    ?/,  =  2y. 

5.  Find  the  co-ordinates  of  the  centre  of  curvature,  and  the 
equation  of  the  evolute,  of  the  hypocycloid  'j^  -\-  y^  —  a^. 

x,=x-{-^  i/^%    y,  =  y  +  3f  :rY;     {x^  +  yj^  +  (x^  -  y^f 
—  2al 

6.  Find  the  evolute  of  the  equilateral  hyperbola  xy  =  m'. 

Note. — First  prove  that 

m (m    ,    xV  ^  m  fm       x  V 

and  thence  derive  the  equation  of  the  evolute. 


DIFFERENTIAL  CALCULUS  AND  PLANE  CURVES.    145 


ENVELOPES. 

169.  Let 

f{x,y,a)  =  0 (1) 

be  the  equation  of  a  curve,  a  being  some  constant  quantity.  If 
we  assign  different  values  to  a,  we  will  obtain  a  series  of  distinct 
curves,  but  all  belonging  to  the  same  system  or  family  of  curves. 
One  of  the  curves  of  this  family  can  be  obtained  by  increasing  a 
by  h,  thus  converting  (1)  into 

f{x,y,a  +  h)=0 (2) 

If  h  be  supposed  indefinitely  small,  the  curves  (1)  and  (2) 
are  said  to  be  consecutive. 

The  points  of  intersection  of  the  curves  (1)  and  (2)  approach 
definite  limiting  positions  as  li  approaches  0,  and  the  locus  of 
these  limiting  positions,  as  different  values  are  assigned  a,  is 
called  the  Envelope  of  the  system  f{x,  y,  a)  =  0. 

The  quantity  a  which  remains  constant  for  any  one  curve  of 
the  series,  but  varies  as  we  pass  from  one  curve  to  another,  is 
called  the  variable  parameter  of  the  series. 

170.  The  envelope  of  a  series  of  curves  is  tangent  to  every 
curve  of  the  sei^ies. 

Let  A,  B,  C  be  any  three  curves  of  the  series,  A  and  B  inter- 
secting at  P,  and  B  and  C  at  P'. 


Fig.  28. 


As  these  curves  approach  coincidence,  the  limiting  positions 
of  P  and  P'  will  be  two  consecutive  points  of  the  envelope  and 
of  the  curve  B.     Hence  the  envelope  touches  B. 

As  an  illustration  see  example  1  under  the  next  article. 


146        DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

171.  To  find  the  equation  of  the  envelope  of  a  given 
series  of  curves. 

The  point  of  intersection  of  (1)  and  (2)  will  be  found  by 
combining  the  equations.  Now,  subtracting  (1)  from  (2),  we 
have 

f{x,  y,  a  +  li)  -f{x,  y,  a)  ^  ^^    ^     ^     ^ 

When  the  curves  approach  coincidence,  li  approaches  0,  and 
(3)  becomes 

d 

j^f{x,  ?/,«)=:  0 .       .       (4) 

Thus,  equations  (1)  and  (4)  determine  the  intersection  of 
any  two  consecutive  curves.  Hence,  by  eliminating  a  between 
(1)  and  (4),  we  shall  obtain  the  equation  of  the  locus  of  these 
intersections,  which  is  the  equation  of  the  envelope. 


EXAVPLES. 

tix 
1.  Find  the  envelope  of  y  ^^^  ax  -\ ,  a  being  the  variable 

parameter. 

y  —  ax  -j-  —   is  the   equation   of   a  line,   as  MN,   Fig.  29. 

When  a  receives  an  increment  li,  the  line  takes  a  new  position, 
say  M  N',  which  intersects  the  former  line  at  c.  As  h 
approaches  0,  c  approaches  p,  a  point  on  the  locus  (APm)  of  all 
similar  intersections. 

Differentiating  with  respect  to  a,  x  and  y  being  constants, 
we  have 


m  ,  ^  ^  y/m^ 


0  =  a; 5 ;      whence     a       _ 

a'  '     X 


y  =  ±\  Vmx  +  Vwx],    or    y^  =  4,mx. 
which  is  the  equation  of  a  parabola. 


DIFFERENTIAL  CALCULUS  AND  PLANE  CURVES.    147 


Let  it  be  observed  that  the  problem  is  the  same  as  that  of 


m 


finding  the  curve  of  which  y  =  ax  -\ is  the  tangent. 

Cv 


Fig.  29. 

2.  Find  the  curve  whose  tangent  is,  y  =  mx  +  a  V'nf  -\-  I,  m 
being  the  variable  parameter.  x'^  +  'if  =  «%  a  circle. 

3.  If  a  right  triangle  varies  in  such  a  manner  that  its  area  is 
constantly  equal  to  c,  find  the  envelope  of  the  hypothenuse,  or 
the  curve  to  which  the  hypothenuse  is  the  tangent. 

Let  OA  =  a,  OB  =  b;  then  the  equation  of  AB  is 


a       0 


(1) 


But  ab  =  2c,  or  b  =  ^ 
a 


a^2c~    ' 


(2) 


where  a  is  the  variable  parameter. 

/2cx 
Differentiating  (2),  we  get  a  =  y  - — ,  which,  substituted 

if 
/J 

(2),  gives  xy  =  — ,  an  equilateral  hyperbola. 


in 


148         DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

Solve  the  preceding  problem  on  the  hypothesis  that  the  hy- 
pothenuse  is  constantly  equal  to  c. 

x^  -^  y   =  c^,  the  hypocycloid. 
Since  a  normal  to  a  curve  is  tangent  to  the  evolute  of  the 
curve,  the  latter  is  the  envelope  of  the  successive  normals,  or  the 
locus  of  their  intersections. 

4.  Find  the  evolute  of  the  parabola  y"^  =  4ax,  taking  the 
equation  to  the  normal  in  the  form  y  =  m{x  —  2a)  —  am'',  m 
being  the  variable  parameter.  27«?/^  =  A{x  —  2ay. 

5.  Find  the  evolute  of  the  ellipse  a^y^  +  ¥x'^  =  a'b'',  taking 
the  equation  of  the  normal  in  the  form 

hy  =  ax  tan  a'  —  {a^  —  If)  sin  a', 

where  the  variable  parameter  a'  is  the  eccentric  angle. 

{axf  +  {hyf  =  {a'  -  b'f.     See  Ex.  2,  Art.  168, 

6.  Find  the  curve  whose  tangent  is  y  =  mx  -{-  {d'm^  -\-  ¥Y, 
m  being  the  variable  parameter.  a'^y^  +  ¥x''  =  a''b'\ 

7.  Find  the  envelope  of  the  family  of  parabolas  whose  equa- 
tion is  y^  =  a{x  —  a).  y  =  ±  {x. 

8.  Find  the  locus  of  the  intersections  of  a;  cos  a  -\-  y  sin  a  =  j^ 
with  itself  as  a  increases  continuously.  x^  +  ?/  =  P^- 

9.  Find  the  envelope  of  all  ellipses  having  a  common  area 
(ttc^),  the  axes  being  coincident.  xy  =  ±  ^c''. 

10.  Find  the  evolute  of  the  curve  x^  +  y^  =  a^,  the  equation 
of  whose  normal  is  y  cos  «'  —  x  sin  a'  =  a  cos  2a',  where  a'  is  the 
angle  which  the  normal  makes  with  the  axis  of  x. 

(a;  +  y)i  -j-{x  -  yf  =  2al     See  Ex.  5,  Art.  168. 

11.  Find  the  equation  of  the  curve,  the  equation  of  its  tan- 
gent being  y  =  2mx  -f-  in\  where  m  is  the  variable  parameter. 

iiyy  +  ii^r  =  0. 

TRACING  CURVES. 

172.  The  Eudimentary  Method  of  tracing  a  curve  is  to 
reduce  its  equation  to  the  form  of  y  =  f{x) ;  that  is,  solve  the 


DIFFERENTIAL   CALCULUS  AND  PLANE  CURVES.    149 

equation /(a;,  y)  =i  0  for  y,  assign  values  to  x,  find  the  correspond- 
ing values  of  y,  draw  a  curve  through  the  points  thus  determined, 
and  it  will  be  approximately  the  curve  required.  This  process 
is  laborious,  and  often  impossible  on  account  of  our  inability  to 
solve  f{x,  y)  =0  for  y. 

The  General  Form  of  a  curve  is  usually  all  that  is  desired, 
and  this  can  generally  be  found  by  determining  its  singular  or 
characteristic  points  and  properties,  and  these  are  embraced 
chiefly  in  the  position  of  certain  turning-points  of  the  curve, 
the  direction  of  curvature  between  these  points,  and  where  and 
how  the  branches  intersect  or  meet  each  other.  In  addition,  we 
may  find,  by  previous  methods,  where  the  curve  cuts  the  axes, 
whether  or  not  it  has  infinite  branches,  asymptotes,  etc. 

173.  Direction  of  Curvature.  —  The  terms  Convex  and 
Concave  have  their  ordinary  meaning  when  applied  to  the  arcs 

of  curves. 

Thus,  as  seen  from  some  point 
below,  the  arcs  AB^  and  UD  are 
concave,  and  B^  0  and  DB  convex. 
174.  A  Point  of  Inflection 
is  the  point  at  which  the  curve 
changes  from  concave  to  convex, 
or  from  convex  to  concave ;  as  the 
points  B^,  C,  D. 

Principles. — The  slope  (-^-1  of  the  curve  evidently  de- 
creases as  the  point  P{x,  y)  moves  from  A  along  the  curve  to 
the  right  until  P  reaches  B^ ,  and  then  increases  until  P  reaches 

dv 
C,  etc.     Therefore  (1)  when  the  arc  is  concave,  -rj—  decreases  as 
'  ax 

(Til 
X  increases,  hence  (Art.  25)  its  derivative  -^    is   — ;    (2)   when 

^-  .  dy  .  .  ,  d'y  . 

the  arc  is  convex,  -i^  increases  as  x  increases,  hence  -3-5  is  +. 
'  dx  dx" 

Therefore,  I.  At  any  point  of  the  curve  y  —  f{x),  the  curve 
is  concave  or  convex  according  as  -^4  is  negative  or  positive. 


150        DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

II.  The  roots  of  -r^  =  0  or  oo  which  will  render  -^  a  maxi- 
ax^  ax 

mum  or  minimum  are  the  abscissas  of  the  points  of  inflections. 


EXAMPLES. 

Examine  the  following  curves  for  concave  and  convex  arcs, 
and  for  points  of  inflection. 


Since  ^-r^  is  +,  the  curve  is  convex  at  every  point. 

2.  y  =  x'  -  6a:'  +  17a;  -  6.  ^  =  6(a;  -  2). 

The  root  of  a;  —  2  =  0  is  2,  the  point  of  inflection ;  the  curve 
is  concave  when  x  <  2,  convex  when  a;  >  2. 

3.  y  =  x'  —  12a;'  +  48a;'  -  50. 

Points  of  inflection,  a;  =  2,  a;  =  4;  curve  convex  when  a;  <  2 
and  >  4,  and.  concave  when  2  <  a;  <  4. 

_  a;  — 2  dy 2 

^-  y  -  x-^'  dx-^  ~  (a; -3)=* 


Point  of  inflection  at  a;  =  3;  convex  when  a;  >  3,  concave 
when  a;  <  3. 

/        -v  d'^V  1 

5.  .y  =  log(a;-l).  _  =  _^__ 

-  =  00  gives  {x  —  1)'  =  0,  which  has  two  equal  roots; 


(a;-l) 
hence.  Art.  140,  there  is  no  point  of  inflection;  curve  concave 


DIFFERENTIAL  CALCULUS  AND  PLANE  CURVES.    151 


6.  Prove  that  the  curve  y  = 


has  points  of  inflection 


at  (0,  0),  (a  V^,  fa  VI),  {- a  VI,  -  la  4/3). 

7.  Prove  that  the  witch  of  Agnesi,  x^y  =  4a'(2a  —  y),  has 
points  of  inflection  at  (±  %a  VS,  !«),  and  is  concave  between 
these  points  and  convex  outside  of  them. 

8,  Find  the  points  of  inflection  ot  y  =  sin  2x  +  cos  2x. 


SINGULAR  POINTS. 


175.  The  Singular  Points  of  a  curve  are  the  turns  and 
multiple  points. 

A  Turn  in  rectangular  co-ordinates  is  a  point  at  which  a 
curve  ceases  to  go  (1)  up  or  down,  or  (2)  to  the  right  or  left, 


Fig.  31. 


and  begins  to  go  in  the  opposite  direction.  The  former,  as 
B,  E,  F,  G,  are  called  y-turns,  and  the  latter,  as  C,  D,  H, 
x-turns. 

The  x-tiirns  and  y-turns  evidently  occur  at  the  maximum  or 
miniinum  values  ofx  and  y,  respiectively . 


152         DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

A  Multiple  Point*  is  one  through  which  two  or  more 
branches  of  a  curve  pass,  or  at  which  they  meet.  A  multiple 
point  is  double  when  there  are  only  two  branches;  triple  when 
only  three,  and  so  on. 

A  Multiple  Point  of  Intersection  is  a  multiple  point  at 
which  the  branches  intersect  (Fig.  32,  a). 


c 
Fig.  32. 


An  Osculating  Point  is  a  multiple  point  through  which  two 
branches  pass,  and  at  which  they  are  tangent  (Fig.  32^  b,  c). 

A  Cusp  is  a  multiple  point  at  which  two  branches  terminate 
and  are  tangent  (Fig.  32,  d,  e).  A  cusp  or  osculating  point  is 
of  the  first  or  second  species  according  as  the  two  branches  are 
on  opposite  sides  (Fig.  33,  h,  d)  or  the  same  side  (Fig.  32,  c,  e) 
of  their  common  tangent. 

A  Conjugate  Point  is  one  that  is  entirely  isolated  from  the 
curve,  and  yet  one  whose  co-ordinates  satisfy  the  equation  of  the 
curve.  _ 

For  example,  in  the  equation  y  =  {a-\-  x)  Vx,  if  x  is  negative 
y  is  imaginary,  yet  the  co-ordinates  of  the  point  [x  =  —  a,  y  ■= 
0)  satisfy  the  equation.  Hence  (—  a,  0)  is  a  conjugate  point. 
A  conjugate  point  is,  generally,  the  intersection  or  point  of 
meeting  of  two  imaginary  branches  of  the  curve,  and  may,  in 
exceptional  cases,  also  lie  on  a  real  branch  of  the  curve. 

There  are  other  singular  points,  such  as  Stop  Points,  at 
which  a  single  branch  of  a  curve  stops  suddenly,  and  Shooting 
Points,  at  which  two  or  more  branches  stop  without  being  tan- 
gent to  each  other.  But  as  these  rarely  occur,  they  are  omitted 
in  this  book. 

*  See  Taylor's  Calculus. 


DIFFERENTIAL   CALCULUS  AND  PLANE  CURVES.    153 

176.  To  determine  the  positions  of  the  singular  points  of 
a  curve. 

Let  u  =f{x,  y)  =  0  be  the  equation  of  the  curve,  free  from, 
radicals.     Then  (Art.  109) 

du 

dy  _        dx 

dx  du 

dy 

(a)  Jjor  the  a^-turns,     -^  =  oo  ;     .-.     -—  z=  0, 
^  dx  '  dy 

(b)  For  the  ?/-turus,     i^  =  0  :     .-.     ^*  =  0. 
^  ^  "^  dx  *  dx 

du 

(c)  For  multiple  points,  -f^,  by  definition,  has  two  or  more 

dit 
values;  hence,  since  u  contains  no  radicals,  -f-  must  be  of  the 

form  ~.     Therefore 

-T-  =  0    and    -r-  =  0. 
dx  dy 

Hence,  to  find  the  :r- turns  we  have  u  —  0  and  - —  =  0:  to 

dy 

find  the  ?/-turns,  we  have  ?<  =  0  and  -— -  =  0;  and  the  values  of 

y  and  x  which  satisfy  all  these  equations  are  the  co-ordinates  of 
the  multiple  points. 

177.  To  determine  the  character  of  the  multiple  points 
of  a  curve. 

From  the  definitions  of  the  multiple  points  it  follows  that: 

dti 

I.  At  a  multiple  point  of  intersection  -j-  has  two  or  more 

unequal  real  values. 

dv 

II.  At  an  osculating  point  or  a  cusp  -7^- has  two  equal  values. 

dv 

III.  At  a  conjugate  point  at  least  two  of  the  values  of  -j-  are 

(.too 

imaginary. 


154        DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 


du 
dtj 


EXAMPLES. 
Find  the  singular  points  of  the  following  curves. 
1.  u  —  x'  —  xy-^if  —  Z  —  ^ (1) 


=  -a^  +  %  =  0; 


du 


m       ^="^x-,j  =  0.     .     .     (3) 


From  (1)  and  (2)  we  find  (2,  1),  (—  2,  —  1),  the  co-ordinates, 
of  the  cc-turns  A  and  A'. 

Y 


Fig.  33. 

From  (I)  and  (3)  we  find  (1,  2),  (—  1,  —  2),  the  co-ordinates 
of  the  ?/-turns  B  and  B\ 

Since  neither  pair  of  these  values  satisfies  (1),  (2),  (3),  the 

curve  has  no  multiple  points. 

2.     u  =  Af  -  (25  -  x"")  (a;'  +  7)  =  0. 

5F  =  «^  =  °' 


du 
dx 


=  2x{x^  +  7)  -  2a:(25  -  x")  =  0. 


Fig.  34. 


From  these  equations  we  find 
{a)  the  .'T-turns,  (5,  0),  (—  5,  0), 

(±1/177,0); 

(b)  the  y-turns;  (3,  ±  8),  (0,  ±^^7), 

(-3,  ±8). 


DIFFERENTIAL   CALCULUS  AND  PLANE  CURVES.    155 


The  figure  (34)  is  only  an  approximate  representation  of  the 
curve. 

a;-turns,  (±  4,  18),     and     (±  ~  |/3l7,  ^). 

?/-turns,  (0,  4),  (0,  0),     and     (±  ^  VvT,  19^1 
0.      .     (1) 
=  a(2a;^  -  3^')  =  0.      .     (2) 

=  4x(a;=  +  mj)  =  0.      .     (3) 


4.       ?i  =  a;*  +  2ax'y  —  ay 

clu 
dy 

du 
dx 

.'.    ^-turns,  (0,  0),  {a,  —a),  {—a,  —a); 


Fig.  35. 


Now  there  appears  to  be  an  a;-tui-n  and  a  y-turn  at  the  point 
(0,  0),  and  in  a  certain  sense  this  is  evidently  true;  but  we  should 
regard  the  result  as  signifying  that  (0,  0)  is  a  multiple  point  of 
S07ne  kind,  since  x  =  0,  y  =  0  satisfy  equations  (1),  (2),  and  (3). 

Let  us  now  determine  the  character  of  the  point.  Dividing 
(3)  by  (2),  we  have 

dy  _  4a;'  +  Aaxy 
dx       2>ay'^  —  2ax^' 

Our  object  now  is  to  find  the  value  of  the  slope  -^  at  the 

dx 

dii 
multiple  point  (0,  0).     For  these  values  of  x  and  y,  -—-  assumes 

the  form  of  -,  hence  the  value  required  may  be  obtained  by 
Art.  137. 


156        DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

We  see  from  Ex.  5,  Art.  137,  that  ^  =  0  and  ±  V2  at  the 

ax 

point  (0,  0).  Hence  the  origin  (0,  0)  is  a  triple  point,  the  three 
branches  which  pass  through  the  point  being  inclined  to  the 
a;-axis  at  the  angles  0,  tan~^  Vs  and  tan"^  (—  ^2),  respectively, 
as  in  the  figure.     See  Art.  179. 

5.  y'^  =  rt V  —  x\ 

2;-turns,  (0,  0),  [a,  0),  (—  a,  0); 


2^-turns,  (0,  0),     |  l/2, 


±la 


-lv%.\. 


The  point  (0,  0)  is  a  double  point  of  intersection,  since  at 

,1    ,       .   ,   dy 

that  point  -p-  =  ±  rt. 

6.  Examine  y'^{(t'^  —  x')  —  x^  =  0  for  multiple  jDoints. 

dv 
At  the  point   (0,  0),    -~  =  ±  0;  that  is,  it  has  two  equal 

values;  hence  (0,  0)  is  an  osculating  point  or  a  cusp;  and  since 
the  curve  is  symmetrical  with  respect  to  both  axes  the  point  is 
evidently  an  osculating  point  of  the  first  species. 

7.  Determine  the  general  form  of  the  curve  y"^  =  n'^x\ 
When  X  =  oo ,  y  =  ±  co  ;  hence  the  curve  has  two  infinite 

branches,  one  in  the  first  and  one  in  the  fourth  quadrant. 

When  X  is  negative,  y  is  imaginary; 
hence  the  curve  does  not  extend  to  the 
left  of  the  y-axis. 

When  X  =  0,  y  =  0;  hence  both 
branches  start  from  the  origin. 

At  the  point  (0,  0),  y^=  ±0;  hence, 

since    the    curve  is   symmetrical  with 
Fig.  36.  respect  to  the  a;-axis,  the   origin   is   a 

cusp  of  the  first  species. 


DIFFERENTIAL   CALCULUS  AND  PLANE  CURVES,    157 


Again,  since   y~-  =  ± 


3a 

2V2 


-.,  the  upper  branch  is  convex  and 


Fig.  37. 


the  lower  concave. 

8.  Examine  the  curve  {y  —  x'y  =  x^,  or  y  =  x^  ±:  x^. 

Has  two  infinite  branches,  one  in  the  first  and  one  in  the 
fourth  quadrant,  both  starting  from  the 
origin.  For  every  positive  value  of  x,  y 
has  two  real  values,  both  of  which  are  posi- 
tive as  long  as  x  <1,  but  at  the  point 
where  ^'  =  1  the  lower  branch  crosses  the 
a;-axis.  The  origin  is  a  cusp  of  the  second 
species. 

178.    Tracing    Polar     Curves.       Let 
f{r,  6*)  =  0  be  the  polar  equation  of  the  curve. 

{a)  By  solving  the  equation  f{a,  d)  =  0  for  6,  we  find  the 
direction  of  the  curve  at  the  point  r  =  a.  It  a  =  0,  the  values 
of  0  will  be  the  angles  at  which  the  curve  cuts  the  polar  axis  at 
the  pole. 

(b)  By  solving  the  equation  -jjr  =  0  for  d  we  find  the  values 

of  6  for  which  r  is  a  maximum  or  minimum,  or  the  r-turns,  at 
which  the  curve  is  perpendicular  to  the  radius  vector. 

9.  Trace  the  curve  ?•  =  a  sin  36*,  Fig.  38. 

(a)  Making  7'  =  0,  we  have  sin  3d  =  0;  hence  6  =  0,  f  tt,  ^tt, 

which  are  the  angles  at  which  the  curve  cuts  the  polar  axis  at 

the  pole. 

cIt 
{b)   -Tn  =  3a  cos  36  =  0;  hence  the  values 


d6 


7C 


Fig.  38. 


of  6  at  the  r-turns  are  \n,  — ,  \7t,  at  which 

points  r  =  «,  —a,  -\-  a,  respectively. 

dv 
Since  ^^  =  3a  cos  06,  r  increases  from  0  to 
du 


a,  while  6  increases  from  0  to  ^tt;  r  decreases  from  «  to  —  a, 
while  6  increases  from  \7i  to  ^ix;  r  increases  from  —  a  to  -{-  a, 
while  6  increases  from  ^rc  to  f  ;r;  and  r  decreases  from  a  to  0, 


158         DIFFFRENTIAL  AND  INTEGRAL   CALCULUS. 

while  0  increases  from  f;r  to  tt.    Further  revohition  of  the 
radius  vector  would  retrace  the  loops  already  found. 

10.  Trace  the  curve  r  =  a  sin  2(9,  Fig.  39. 
From  this  and  the  previous  example,  we 

infer  that  the  locus  ofr  =  a  sin  7id  consists  of 
n  loops  when  n  is  odd,  and  2n  loops  when  n 
is  even. 

11.  Trace  the  curve  r  =■  a  cos  B  cos  26*,  or 
r  =  a(2  cos'  6  -  cos  d),  Fig.  40. 


Fig.  39. 


12.  Trace  the  lemniscate  r^  =  a'  cos  26,  Fig.  41. 


Fig.  40. 


0 

Fig.  41 


179.  The  character  of  multiple  points  in  rectangular  co- 
ordinates may  often  be  more  easily  determined  by  changing  to 
polar  co-ordinates,  and  applying  («)  of  Art.  178. 

Thus,  in  Ex.  4,  Art.  177,  make  ^  =  ?'  sin  6  and  x  —  r  cos  6, 
divide  by  r',  and  we  have 

r  cos'  6  -f  2a  cos'  6^  sin  ^  -  «  sin'  0  =  0. 


Now  making  r  =  0,  and  we  have 

sin  6=0,    and     tan^  6  =  2; 

that  is,  the  angles  at  which  the  curve  cuts  the  a;-axis  at  the 
origin  are  sin~^  0,  tan~^  V2,  tan""^  —  V2. 
Trace  the  following  curves: 

13.  The  Cissoid,  y\2a  -  x)  =  xf. 

14.  The  Conchoid,        x'f  =  {b'  -  f){a  +  ^y. 


DIFFERENTIAL  CALCULUS  AND  PLANE  CURVES.     159 

15.  The  Witch,  {x'  +  4a^)^  =  Sa\ 

16.  The  Lituus,  r  VB  =  a. 

n 

17.  The  Parabola,         r  =  a  sec"  - . 

n 

18.  The  Curve,  r  =  rtsin'-. 

19.  The  Cardioid,         r  =  a{l  -  cos  6). 

20.  The  Hypocycloid,  x^  +  ?/'  =  a^. 

21.  Examine  ay^  =  x^  —  hx^  for  multiple  points. 

(0,  0)  is  a  conjugate  point. 

22.  Prove  that  y^  =  x""  and  (y  —  xY  —  x^  have  cusps  of  the 
first  species  at  the  origin. 

23.  Prove  that  a;^  —  %ax?i)  —  axy^  +  a^if'  =  0  has  a  cusp  of 
the  second  species  at  the  origin. 

24.  Prove  that  x*  —  ax'^y  —  axy'^  +  a^y'^  =  0  has  a  conjugate 
point  at  the  origin. 

25.  Prove  that  the  multiple  point  of  ay^  =  (x  —  a)'\x  —  h) 
at  {a,  0)  is  (1)  a  conjugate  point  if  a  <  J,  (2)  a  double  point  if 
a  >  J,  and  (3)  a  cusp  \i  a  =■  b. 


CHAPTER   VIII. 

GENERAL  DEPENDENT  INTEGRATION. 

FUNDAMENTAL  FORMULAS 

180.  The  differentials  in  the  following  twenty-two  formulas 
are  the  fundamental  integrable  forms,  to  one  of  which  we 
endeavor  to  reduce  every  differential  that  is  to  be  integrated  by 
the  dependent  process  (Art.  51): 

v'^civ  =  ■ — —  +  a 

/dv 
—  =  log  {v)  +  C,    or    log  V  -\-  log  c  =  log  cv. 

3.  fa^'dv  =  ~ 1-  C. 

J  log  a 

4.  J &'dv  =  e^  ^  G. 

5.  /  cos  V  dv  =  sin  z^  +  C. 

6.  /  sin  V  dv  =  —  cos  v  -\-  C. 

7.  /  sec^  V  dv  =  tan  v  -\-  C. 

8.  /  cosec'  vdv  =  —  cot  v  +  C. 

9.  /  sec  V  tan  v  dv  =  sec  y  +  0. 

160 


GENERAL  DEPENDENT  INTEGRATION.  161 

10.  /  cosec  V  cot  V  dv  =  —  cosec  y  +  C. 

11.  /  tan  V  dv  =  log  sec  v  -\-  G. 
13.     /  cot  V  dv  =  log  sin  v  +  (7. 

13.  /  sec  V  dv  =  log  (sec  v  +  tan  «;)  +  O. 

14.  /  cosec  V  dv  =  log  (cosec  v  —  cot  v)  +  C'. 

^^      r     dv  .     ,  ^ 

15.  /     ,  ,-sin-^j;+  (7. 

P    —  dv  ,  ^ 

16.  / =  cos-i  v  +  (7. 

^   i/l  -  v' 

17.  /':r4^  =  tan-i  i^  +  C. 
ty    1  -\-  V 

18-    Z"^*^  =  cot-i  V  +  (7. 
«^    1  +  y" 

19.  f — ^IL^^  =  sec-i  v+0. 

20.  f-^M=r  =  cosec-1  y  +  G. 


v 


Vv'  -  1 


1.     /    , =  vers"^  V  4-  G. 


9  _ 


2.    f—J-^=  =  log  (t;  +  4/y'  ±  m)  +  G. 


\/  v"  ±  m 

In  these  formulas  v  may  be  the  independent  variable,  or 
some  function  of  this  variable,  and  the  process  of  integration 
consists  largely  in  reducing  or  transforming  any  given  differen- 
tial into  one  of  the  above  forms. 


162         DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

REDUCTION  AND  INTEGRATION  OF  DIFFERENTIALS. 

181.  Reduction  of  Differentials  is  the  process  of  reducing 
them  to  integrable  forms,  and  is  effected  chiefly  (1)  by  constant 
multipliers,  (2)  by  decomposing  or  separating  them  into  their 
integrable  parts^  and  (3)  by  substitution. 

182.  Reduction  of  Differentials  by  Constant  Multipliers. 

Principle.     The  value  of  any  differential  of  the  form  c  I  dv 

remains  unchanged  if  dv  be  multiplied  and  c  be  divided  by  the 

same  constant. 

EXAMPLES. 
Find  the  following: 

1.  ?/  =    /  — ■ — -.  Ans.  —-tan  ^  — ^  +  6\ 

We  reduce  this  to  the  form  of  17,  Art.  180;  thus 


=  Answer, 


/ndv  V 

,  (See  15,  Art.  180.)     Ans.  n  sin'^  77^  +  C'. 


/: 


^   n_    r       iv        ^,^       I     )l^ =  Answer. 

V,l^^_^       V.J     ^,_(^J 

Let  the  student  compare  these  results  with  formulas  (13) 
and  (15),  page  65,  and  in  a  similar  manner  deduce  formulas 
(14)  and  (16)  on  that  page. 


GENERAL  DEPENDENT  INTEGRATION.  163 

_  3    P{2x  +  2)dx  _  3    rdix'  +  2x  +  5)  _ 
^~2t/a;^  +  2a;  +  5~2e/     rc^  +  2a;  +  5    ~        " 

(Formula  2,  Art.  180.) 

f  Zx'dx  .   ,       .^  V        ^ 

4-  y  T^+T-  ^T  log  {Ix  +  4)  +  a 

5.  /"      "^"^      .  isin-^-^+C. 
^   4/2 -9a;"  V^2 

6.  f--J^==.  vers-1  3.^  +  a 

g      r_dx__  ^taii-»a;l/5+ a 

■  ^  1  +  5.1;*'  4^ 

^   2  +  3.1;''  4/6  ^ 


11.  f    ,— _.  -i  sec-^  a;|/f  +  a 
•^  a;|/2:^;'^-3  4^ 

12.  /     .-  I  vers  ^  —  +  (7. 

13.  f  -^^  .  4-- cos-i^i^+ a 

^   V3  -  2a;^  4/2 


164         DIFFERENTIAL  AND  INTEGRAL   CALCULUS, 
bdx 


15.    / ^^^  Vb  sec-i  xV'i  +  G. 


16.    f        ^^^  l\^&Qc''xV^-\-C. 

Some  of  the  preceding  examples  may  be  conveniently  solved 
by  formulas  (19)  and  (21),  page  65. 


REDUCTION   OF  DIFFERENTIALS   BY  DECOMPOSITION. 

183.  The  process  of  reducing  differentials  to  integrable 
forms  consists  largely  in  separating  them  into  their  integrable 
parts. 

184.  Elementary  Diflferentials.  In  these  the  necessary 
reductions  are  effected  by  the  simple  operations  of  algebra. 

EXAMPLES. 
Find  the  following: 

1.  /^^^^^-  t  log  (4^;^  +  1)  +  I  tan-i  {2x)  +  C. 

f{Zx  +  5)      _    r   ^xdx  p     bdx 

r{x-Z)  ^^  -  (1  -  x"")"  -  3  sin-1  xi-C. 


Vl  -x' 

(2  -5a 

|/4«  -  2a;'""^"  4^2  ^~"'       '" '         V2 


I    f-^=J^dx.  -4- (2a;  -  x'f  -  -t  vers-^  x  +  C. 

O       a/ At  _  9.^^  VO  a/O. 


r  Vx'  -  1   .. ,  |/;r^^^_     a;°  -  1     "I 

(a;"  -  1)*  +'  cosec-^  x+  C 


GENERAL  DEPENDENT  INTEGRATION. 


165 


/Va  +  X 
Va-x 


dx. 


Va  -{-  X  _     a-\-  X 


Va  —  X       Va"  —  X- 


•] 


x-{-  1\* 


8.    f^dx. 


sin-^--Va'-x^+C. 
a 


~  +  ~+x+G. 


\-\-^\-^-^^og{i-Yx)  +  a 


185.  Trigonometric   Differentials. —  In  reducing  these  we 
use  the  elementary  formulas  of  Trigonometry,  such  as 

sin"  X  -\-  cos"  X  =  1,     sin  2a;  =  2  sin  x  cos  a;, 
cos  2iX  =  cos°  X  —  sin*  x,  etc. 


EXAMPLES. 


Find: 
1.     /  sin'  x  dx. 


—  cos  X  -\-^  cos'  X  -\-  G. 
sin'  a;  =  sin  x  {\  —  cos'  a;)  =  sin  a;  —  (cos'  x)  sin  tc. 

/  (sin'a;)«Ja;=  J  sinxdx  -\-  J  (cosxyd{cosx). 

2.  I  sin^  X  dx.        =    /  (1  —  cos'  a;)'  sin  x  dx. 

—  cos  «  +  f  cos"  X  —  \  cos^  a;  +  C'. 

3.  /  sin^  x  ^a;.  —  cos  x  +  cos'  x  —  \  cos"  a;  +  t  cos'  a;  +  C'. 

4.  /  cos'  .r  Ja;.  sin  a;  —  -J  sin'  x  -\-  C. 

cos'  a;  =  cos  a:(l  —  sin'  x). 


166         DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

5.  /  COS"  X  dx,  sin  x  —  ^  sin'  x -\-  \  sin^  x  -{-  C. 

In  like  manner    /  cos™  x  dx  and    /  sin™  x  dx  can  be  found 
•where  m  is  any  odd  positive  integer. 

6.  /  sin*  X  cos'  x  dx.  \  sin^  ^  —  \  sin'  x  -{-  C. 

sin*  X  cos'  X  =  sin*  a:(l  —  siu^  x)  cos  a;. 

In  a  similar  manner    /  sin™  x  cos"  x  dx  may  be  found  when 
either  m  or  7i  is  any  odd  positive  integer. 

7.  /  sin"  X  cos*  x  dx.  —  \  cos'  x  -\-\  cos'  x -\-  G. 

8.  I  &m^  X  Qo^"  X  dx.  -^  sin' a;  —  I  sin' a;  +  I  sin' a; 

—  \  sin'  a;  +  C'. 

9.  /  sin'  X  Vcos  a;  dx.  —  |(cos  a;)^  +  f(cos  xf  +  (7. 

10.  /  cos'  xdx.  i"  +  ^  ^^'^  ^^  "^  ^° 

cos'  a;  =  ^  +  i  cos  2x. 

11.  /  sin'  xdx.  o  ""  i  s"^  ^^  +  ^> 

or  r —  ^  sin  a;  cos  x-\-  G. 

12.  /\     ^^      .     r=z   /"'-^^c/^^.l  logtanjc+a 
t'   sm  X  cos  a;      L     *^    tan  a;      J 

13.  /  -T-7, r-'  tan  a;  —  cot  a;  +  C. 

«/  sm  X  cos  a; 

1  sin'  X  +  cos''  a; 


GENERAL  DEPENDENT  INTEGRATION.  167 

14,     /  - — ^ .  sec  X  +  cos  X  +  6. 

o      cos  a; 

/siii^  X 
— -. — dx.  4  tau"  x-\- 1  tan'  x  -}-  G. 

cos  a;  ^  ^ 

— ~  —  tan''  a;  sec*  x  =  tan'  a;(l  +  tan^  x)  sec^  a;, 
cos'  a;  V     1  ; 

In  like  manner  — - — dx  or  -. dx  may  be  integrated  when- 

cos   X  sm**  X 

ever  m  —  n  is  even  and  negative. 

/dx 
- — '— .  tan  X  -{-  i  tan^  x  4-  C. 

CDS'*  a; 

17,  /"f^tji^.  -icofx+C. 

^      sm  a; 

186.  Trigonometric  differentials  can  often  be  more  conven- 
iently integrated  as  indicated  by  the  following  solutions. 

18.  /  cos*  x  sin^  x  dx.  —  \  cos^  ^  +  t  ^^s'  x  +  C. 

dy 


Make  cos  a;  =  y;  then  sin  a;  =  Vl  —  ?/^  and  dx  =  — 


.'.    J  cos*  X  sin"  xdx  =  J  —  ?/*(l  —  ^^)f/j/  =  —  ^/y"  +  \y''  +  C"- 

19.     /  tan=  X  dx.  ^  tan*  x  —  ^  tan''  a;  +  log  (sec  a;)  +  C. 

Make  tan  a;  =  y;  then  (?a;  =       •    , . 

2/+1 

.-.     /tan^  a;  ^a;  =S:^ly  =  /(^/^  -  .7  +  ^J^^^ 

=  i!/^-i2/^  +  i  log  (/  +  !)  + a 

This  method  combined  with  that  of  Arts.  212  to  215  affords 
a  complete  solution  of  rational  trigonometric  differentials. 


168         DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

187.  Rational  Fractions.  A  fraction  whose  terms  involve 
only  a  finite  number  of  positive  and  integral  powers  of  the 
variable  is  called  a  Rational  Fraction;  as 

X'  -  2x'  -  13a;'  +  17  , 
^  X  -\-^x  -\-'Z 

To  separate  this  fraction  into  its  integrable  parts,  we  first 
divide  the  numerator  by  the  denominator  and  obtain 

dy  =  {x  —  5x)ax  H r-r-77 — r-K«^- 

^       ^  '  a;^  +  3a;  +  2 

Again,  by  separating  the  fractional  part  of  this  quotient  into 
two  parts  (its  "partial"  fractions),  we  obtain 

10a;  +  17  '  ,  7^a;     ,     Mx 

-dx  =  — -—  4- 


a;'  +  3a:  +  2  a;  +  l'a;  +  2 

dy  =  (a;"  —  5x)dx  -\ -—  + 


a;  +  1    '  a;  +  2' 

dx 


y  =  f{x^  -  5x)dx  +  7/^^  +  3/ 


+  1    '     »/    a;  +  2  ' 

x^        ^x^ 
or  2/  =  --  — +  71og(a;+l)  +  31og(a;+2)+67. 

That  is,      y  =  g  +  log  {x  +  l)-(a;  +  2)^  +  C. 

The  first  step  in  the  above  and  similar  operations  is  very 
simple,  and  it  is  our  present  purpose  to  show  how  the  second 
step,  the  separation  and  integration  of  fractions  whose  denomi- 
nators contain  a  higher  power  of  x  than  the  numerator,  may  be 
effected ;  and  to  render  the  process  as  simple  as  possible  we  shall 
apply  it  to  particular  examples  in  each  of  the  four  cases  that 
may  occur. 


GENERAL  DEPENDENT  INTEGRATION.  169 

188.  Case  I.   Wlien  the  simple  fad  or  s  of  the  denominator 
are  real  and  unequal. 

EXAMPLES. 

1.  Integrate  ^^^  =  ^"^^I^fe^- 

The  roots  of  x""  +  6a;'  +  Sx  =  0,  are  0,  —  3,  and  —  4;  hence 
the  factors  of  x^  -\-  Qx^  +  8^  ^.re  x,  x  +  2,  and  x  +  4. 

.  x  +  \  A    ^       B     ^       C 

^^^""^^     x^  +  6a;-  +  8^-   ^  ^  +  ^^2  +  rTM'       *     *     ^^^ 

Clearing  (1)  of  fractions,  we  have 

a;  +  1  =  A{:x  +  2){x  +  4)  +  B{x){x  +  4)  +  C{x){x  +  2),     (2) 
or    x-\-l={A  +  B+  C)x'  +  (6^  +  45  +  2C)x  +  8J. 

Equating  the  coefficients  of  the  like  powers  of  x,  we  have 

A+B-{-C=0,    6.4+45+2C=l,    8^  =  1. 

Solving  these  equations,  we  find  A  =  },  B  =  I,  and  C  =  —§. 
Substituting  these  values  in  (1),  we  have 

x-\-l  1,1  3 


x'  +  6x'  -{-8x       8x  '  4(a;  +  2)       8{x  +  4)* 
,        dx  ,         dx  ^dx 


8x   '   4(a;  +  2)       8(a;+4)' 
dx    ,    ,    p   dx         o   r   dx 


=  -|loga;  +  f  log  (a;  +  2)  -f  log  (a;  +  4)  +  ^  log  c. 
,      l/cx{x  +  2)^ 


170         DIFFERENTIAL  AND  ISTEORAL   CALCULUS. 

The  values  of  A,  B,  and  C  may  be  obtained  from  (2),  thus  : 
Making  x  =       0,  we  have       1  =       8^;     .•.  A  =       |. 
Making  x  =  —  2,  yfe  have  —  1  =  —  4:B;     .'.  B  =       i. 
Making  x  =  —  4,  we  have  —  3  =       80;     .:  C  =  —  %. 

Principle.   In  this  case,  to  every  factor  of  the  denomiiiator ,  as 

J. 
X  —  a,  there  corresponds  a  partial  fraction  of  the  form  ^  _  ^ 

Find  the  following: 

4,  f  (f-n)'^;_.         log  [(^  - 1)>  +  2r4 

5.   /-(EiJLllIi*:.         log  yx{^  -  2)-(.^  +  3)-  +  a 

J     x"  ^  X  —  m 

nx^J^of-8,  X'X^  X\X-2Y     ,     ^ 

189.  Case  II.   Tf7^e7^  some  of  the  simple  factors  of  the  de- 
nominator are  real  and  equal. 

EXAMPLES. 

^  ix"^  +  x)dx 

1.  Integrate  dy  =  ^^  _  ^y^^  _  ^y 

:)f  +  x                  A         .       B      ^       C 
Assume   -, kz^tz T\  —  7Z ovi  +  ;:: o  + 


(x-2y{x-l)       {x-2y~^  x-2~^-x-l' 
Clearing  of  fractions,  we  have 

X'  +  X  =  A{x  -  1)  +  S{x  -  2){x  -  1)  +  C{x  -  2y 


GENERAL  DEPENDENT  INTEGRATION.  171 

Making  x  =  2,  we  have  Q  =  A;  .\  A  =^  Q. 
Making  a;  =  1,  we  have  2,  =  C;  .'.0  =  2. 
Making  x  =  0,  we  have  0  =  —  .4  +  25  +  4(7 ;     .-.  B  =  —  1. 

•  •     (x-  2y{x  - 1)     {x-2y     x-2'^x-r 

-,  6dx  dx      ,      2dx 

•••     ^y  =  Tz — ^.  -  z — 5  + 


{x-2y      x-2   '   x  —  V 


.  r    dx         r  ^^    ,  c^  r  dx 


log  (a;  -  2)  +  2  log  {x-l)  +  C 


x  —  2 

,    {x-iy       6     ,  „ 

Principle.     /;?  ^//is  case,  to  every  factor  of  the  form  {x  —  «)" 
//iere  corres2}onds  a  series  of  n  partial  fractions  of  the  form 

A  B  K 

{x  —  aY '    (a;  -  «)"""  *  *  '  a;  —  a* 
Find  the  following: 


-s 


{X  -  3)= 

(Qa;'  +  9a;  —  128)  (Za; 
(a;  -  3)Xa:  +  1)    * 
5 


a;  —  3 


17  log  {x  -  3)  -  8  log  (a;  +  1)  +  C. 


^      r  ^^^  1       p  +  Sy  _         2a; +  5 ^ 

^'  ./  (a;  +  2)Xa;  +  3r'  ^^^  ^  +  2/        (a;  +  2)(a;+ 3)  "^ '^• 

,      /'3a; +2,  4a; +3      ,^,      /^^\,/7 


172         DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

190.  Case  III.  Wlie^i  some  of  the  simple  factors  of  the  de- 
nominator are  imaginary  and  unequal. 

EXAMPLES. 

1.  Integrate         dy  =  ^^  ^  ^^^^.  ^  ^y 

Here  the  two  simple  factors  of  x^  -\-  4  are  a;  +  2  i/  —  1  and 
a;  _  2  |/  —  1 ;  we  may  take  these  factors  and  proceed  as  in  Case 
I,  but  the  integrals  obtained  would  involve  the  logarithms  of 
imaginaries;   to  obviate  this,  we  assume 

X  _      A        Bx+C 


(a;  +  l)(a;'+  4)      a;  +  1  '     a;^  +  4 
Clearing  of  fractions,  we  have 

x  =  A{x-'-\-4)  +  {Bx+C){x  +  l).     .     .     .     (1) 
Differentiating  (1),  we  have 

1  =  2.4a;  +  B{x +  1)  +  Ex  +  C.     .'    .     .     .     (2) 
In  (1)  making  x  =  —  1,   we  have   A  =  —  ^. 
In  (1)  making  x  =       0,   we  have    C  =  —  4^  =  +  f . 
In  (2)  making  x  =       0,   we  have   B  =  1  —    C  =  ^. 


••■^^  =  -<^)+<^4-> 


dx    \   ,    J  xdx  \   ,    J    dx 


+  iL-:r— J  +  l 


-       '^[x-^lj   '    'W  +  4:I   '    na;^  +  4y 
...     2/  =  _  t  log  {x+l)  +  j\  log  {x^  +  4)  +  f  tan-^  |-  +  C 

Principle.     In  this  case,  to  every  factor  of  the  denominator 
of  theforin  {x  —  ay  +  h-  there  corresponds  a  partial  fraction  of 

,,      r  Ax  +  B 

the  form  -. Tr~r~n- 

'  {x  —  ay  +  &' 


OENEBAL  BEPENBEWT  INTEGRATION.  173 

*•  y  3M^^-  *  '"S  ijrrpj  +  —  tan-  —  +  C. 

»•  /l^/-  -  +  *  log  ^  -  ^rtan-.  -|  +  a 

6.  /  J^.         JL  ,„g  -4±£J^1  +  ^  tan-  ^  +  a 

191.  Case  IV.    Wlien   some   of  the   simple  factors  of  tJie 
denominator  are  imaginary  and  equal. 


1.  Integrate        dy 


EXAMPLE. 
dx 


1  Ax  +  B    ,    Cx  +  D  ,      E 

Assume     .  ^  ,   q\2/ tt  —  i  2  ,   o\a  + 


(a;^  +  %)\x  -  1)       (a;^  +  3)^   '    a;^  +  3     'a;  -  1* 
Clearing  of  fractions,  we  have 
1  =  (via;  +  B){x-\)  +  {Cx  +  D){x'  +  3)(a;-l)+^(a;^4-3)^     (1) 
Whence        A= -\,     B  =  -  i,     0=  -  j\, 

^  —  16?       ^  —    16* 

,_      1     X  -\-\  l(a;  +  l)l      dx 

11  la; 

•*•    y  =  -^r^-r^x  —  tTTT^og  (x''  +  3) -tan"*  --= 


+  ^log(.-l)-l/ 


dx 


16  ^^  >       4t/   (a;^  +  3)^* 

doc 
The  integration  of  differentials  of  the  form  -. —  ^   '   ,,     may 
^  («a;'  +  J)™        ^ 

be  more  conveniently  obtained  by  Art.  211  or  315. 


174         DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

REDUCTION  BY   SUBSTITUTION. 

192.  Irrational  Differentials.  To  integrate  an  irrational 
differential  which  is  not  of  one  of  the  known  integrable  forms, 
we  first  rationalize  it,  and  then  proceed  according  to  the  previ- 
ous methods. 

To  show,  in  a  simple  manner,  how  rationalization  is  to  be 
effected,  we  shall  apply  the  process  to  a  few  particular  examples. 

EXAMPLES. 
Find  the  following: 

(2  Vx^  i)dx 


^'y  =  f-. 


2Vx{x-\-  nVx  +  5)* 
Make  x  =  z^;     .:  dx  =  2z  dz. 

y  =flX'VlT+^)  = '"« <^'  +  3^  +  ^)  =  i°g  (- + 3  ♦'5  +  5). 

o       /"(2  Vx  +  l)dx 


.  /~V. — 'TF'  ■i/x-{-Vx-{-a 

^y  x""  -\-xvx  ^  ' 

3.    /'  ,  f^    3..  tan-^Vx  +  a 
^  2{x^  +  x') 


4. 


Z'      _^%^^.     [Make  x  =  z'A  sin-^  \/x  +  O. 


3f:r^Vl_a;S 


5.  /  — =7 ^.  log  \- -]  4-  G. 

6.  /    ^   ^   ^.     [Make  x  —  2;°,] 
^   a;*  +  a;^  _  _ 

2  Va;  -  3  f  a;  +  6  f^  -  6  log  (1  +  '^^a;  )  -f  a 
/*    (a;-l)Ja;  .  4^-  2     .    ^ 


GENERAL  DEPENDENT  INTEOBATION.  175 

193.  When  a  +  hx  is  the  only  part  having  a  fractional 
exponent. 

Assume  a  -{-Ix^  «",  where  n  is  the  least  common  multiple 
of  the  denominators  of  all  the  fractional  exponents;  then  the 
values  of  x,  dx,  and  each  of  the  surds,  will  be  rational  in  terms 
of  z. 

EXAMPLES. 

Find 


I.  y  =    Cx""  Vl  +  xdx. 


Assume     \^x  —  z''\         then      Vl -\- x  =  z.      .     .  .  (1) 

Also                        x  =  z'  -1,                       x'  =  (z'  -  ly,  .  (2) 

dx  =  2z  dz.     .  .  (3) 
Multiplying  (1),  (2)  and  (3)  together,  we  have 

J'x'  Vr^'  dx  =f2z\z"-  —  lydz 

=  ¥  -  ¥'  +  1^'  +  ^ 

=  f  (1 +2;)' -  1(1  +  .0^  +  1(1  +  a;)^+ (7. 


p     xdx  2{x  -2)4/1  -\-x 


3.  f-f-^=.  Iog(^L±J^]  +  a 

4.  Jx{a  +  xfdx.  -^^(a  +  xf{^x  -  Za)  +  C. 


194.  When    \'a  +  6a;  +  a;'   or    \/ a  +  Ix  —  x"   is  the  only 
surd  involved. 


A  differential  containing  no  surd  except  Va  -(-  bx  -\-  x'  can 
be  rationalized  by  assuming  Va-^-hx-l- x^  =:  z  —  x',  and  one 
containing  no  surd  except  Vci  +  bx  —  x'  can  be  rationalized  by 


176         DIFFEBENTIAL  AND  INTEGRAL   CALCULUS. 


assuraiug  Vci-\-hx  —  x"  =  {x  —  r)z,  where  r  is  one  of  the  roots 
of  a  +  5x  —  a;''  =  0. 

The  process  is  illustrated  in  integrating  the  following  im- 
portant differentials  (see  Ex.  30,  31,  page  67). 


1.  Find  y  —  f- 


dx 


Va-\-'bx-\-  x^ 

Assume  V a  -\-  hx  -\-  x^  =  z  —  x;  then 

z^  —  a 


a  -{-  bx  —  z^  —  2zx,     X  = 


2z-\-b' 


2{z^  +  bz  +  a)dz 


1  2z  +  h 


(3) 


Va  +  bx-^x'      z'  -\-bz-\-a 

^^^'''^    Va  +  bx  +  x^~^   2.  +  * 

=  log  {2z  +  6)  +  C. 

f-, ~ =  log  {2x  +  b  +2  Va+bx+x^)  +  C. 

^    \'a-\-bx-\-x 


When  ^>  =  0, 


^     ya-\-x 


2.  Find  V  —  f ' 


dx 


Va  -\~  bx  —  x^ 


Eepresent   the  factors   of  a  -\-  bx  —  x'^  =  0   by  x  —  r    and 
r'  —  X,  and  assume 


Va  -\-bx  —  x"  =  V{x  —  r){r'  —  x)  =  {x  —  9')i 


GENERAL  DEPENDENT  INTEGRATION.  'ill 


then  r'  —  X  =  {x  —  r)z'^,        x 


r  +  1' 


_  2(r  -  r')zdz  1 z'  +  1 

"      {z'  +  1)^    '   •  •  ^  )       4/«  +  bx-  X'  ~  {r'  -  r)z 


dx  ^   r    dz 


(1)  X  (2),  f  '^^ ==  =  -  2  / 


|/«  +  Ja:  -  a;=  J   1  ^  z" 

2  tan-i ;?  +  a 


,.  r     _J^ =  _  2  tan-  \/':^-=^''-  +  (7. 

^    |/a  +  6.1;  -  a;'  ^'  "  ^" 

When  6  =  0,     r  =  +  \/a,     and     r'  =  —  Va,     we  have 


iiL__  _of..-i.   h^a±x 


tan-U/     ,_^-"  +C'- 


4/a  —  x'  V      1/fl  —  X 


p dx_ J_lo    /  ^"^  +  2a;  -  |/2  -  aA 

J  X  i^2  +  x-  x''         V2         \  1/2  4-  2a;  +  V^^^x) 


+  G. 


x 


Assume      l/2  +  a;  -  a;=  =  V(2  -  a;)(l  +  x)  =  (2  -  a;)^;  then 

/^  ^a;  _    /"     2^2     _  J_        ;=!  V2-  1 

'^  a;  |/2  +  a;  -"^'  ~  t/  2^'  -  1  ~  4/3    ^^  ^  ^3  ^  1  * 

/dx  

xV^+^^^l       2tan-Ma;  +  |/:^^  +  3:r-l)  +  a 

/^  x'dx       _  .   _j^-l        (a;  +  3)  VS  +  2a;  -  a;' 

•  J    |/3  +  2a;-a;-^'       ^  ''''     "^  2  "' 


178         DIFFEBENTIAL  AND  INTEGRAL   CALCULUS. 

195.  Binomial  Differentials.     Differentials  of  the  form 

r 

x"'{a  +  bx^ydx, 

where  m,  n,  r,  and  s  represent  any  positive  or  negative  integers, 
are  called  binomial  differentials. 

196.  To  determine  the  conditions  under  which  a  binomial 
differential  is  integrable. 

v 

I.  When  -  is  a  positive  integer  the  binomial  factor  can  be 

developed  in  a  finite  number  of  terms,  and  the  differential  exactly 

integrated;  and  when  —  is  a  negative  integer  the  differential  is 

a  rational  fraction  whose  integral  can  be  obtained  by  the  method 
of  Art.  187,  212,  214,  or  215. 

r 

II.  Assume      a  -\-hx^  =  z^\     .'.  {a  +  fe")s  =  z^,      .     .     (1) 

1  m 

and  ,,  =  l_,.-.(!l^«)="',, (3) 

Multiplying  (1),  (2)  and  (3)  together,  we  have 

m  +  l  _j 


x-^{a  +  ix-ysdx  =  ^^z^-  +  '''[-^—j  dz.  ,     .     (4) 

The  second  member  of  (4),  and  therefore  the  first,  is  in- 
tegrable when  ^       -  is  a  positive  or  negative  integer,  by  Case  I. 
^  n 

III.  Assume     a  +  bx^  =  z'x^ ;     . • .  a;™  =  a{z'  -  b)-\ 

■y  1  mm 

X  =  d^{z'  -  iy~%         x^  =  a^{z'  -  ly  %....(!) 
a  +  5a;"  =  -^,       {a  +  lx-)S  =  (fs{z'  -  bf  h^dz,    ...     (2) 

Z^  —  0 


GENERAL  DEPENDENT  INTEGRATION.  179 

s    -  -  i  -1 

and  dx= a^'z^-Hz^  —  5)    »    dz.     (3) 

Multiplying  (1),  (2)  and  (3)  together,  we  have 


s 


m  +  1       r  /m  +  lr 


x'^{a  +  Ix^fdx  = «  "       '^{z'  —  hy^  '»    ^^     V  +  ^-W^.    (4) 

By  Case  I,  the   second   member  of   (4)  is  integrable  when 

771  -\-  1         T 

1-  —  is  a  positive  or  negative  integer. 

Jv  s 

r 

Hence,  x"\a  +  hx^^dx  can  be  integrated  by  rationalization: 
I.  Wlien is  an  integer  or  0,  ly  assuming  a  -\-  Ix"^  =  z^. 

11.    Whe?i 1 —  is  an  integer  or  0,  hy  assuming 

a  -\-  hx'^  —  z^x"-. 
When  the  differential  reduces  to  a  rational  fraction,  which  is 
the  case  when [-  1  is  a  negative,  or [ [-la posi- 

"^t  /IS 

tive,  integer,  it  is  less  laborious  to  integrate  by  a  method  to  be 
subsequently  given. 

EXAMPLES. 

1.  Find     fx'il-^x'fdx. 

^^       m  -\-  1       5  -{-  1       ^ 

Mere =  — - —  =  3,  an  integer,  and  s  =  2;  hence  we 

assume 

l  +  x^  =  z'',  .■.{i  +  x'f  =  z,  .     .     .     .     (1) 

x'  =  z''  -  1,  x'  =  {z'  +  1)' (2) 

Differentiating  (2), 

Qx'dx  =  6{z^  +  lyz  dz (3) 


180         DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 
Multiplying  (1)  and  (3)  and  dividing  by  6,  we  have 

=  f{z'-\-  2z'  +  z')dz 
=  \z'  +  P'  +  K  +  C 
=  1(1  +xf  + 1(1  +  x^f  +  i(l  +  ^'f-+  0. 
rind: 

2.fx^{l  +  ^l'dx.  (l  +  .f(^)  +  a 

3.  rx\l-\-x'fdx. 

A(l  +  a;^)"  -  1(1  +  x^f  +  ^(1  +  x^f  +  C. 

4.  /^^l  +  ^^)-^^^-.  (1  +  ^^)*(^--)  +  ^• 


/3  3^:;  -|-  1      1/7 


Here 

'^±2  _  z:l±i  =  _  1-;    and  '?i  +  -l+r=-i-|  =  -8, 

^      ~        2  3'  M       '  5  2      2 

an  integer;  hence  we  assume 

a 

XJ^x^=x^z^;     .-.  (l  +  a;r^  =  (-^-=^;     •     .     .     (1) 


X' 


-^  =  z'-l;    .     .     .     .     (2) 


dx=r--^^'^..    .     .     (3) 
{z^  -  1)^ 


GENERAL  DEPENDENT  INTEGRATION  181 

Multiplying  (1),  (2),  and  (3)  together,  we  have 

(        1^ 


where  z  =  --/l  + 


X 


X' 


6.    fil  +  xYhlx.     [m  =  0.]  ^        4- a 

^  Vl-\-  x' 

7.  /.-(I  -  2xr'cix.  -  (1  +  4^;)a  -  ^-'f  +  c. 

8.    fx-Ha  +  xT^^dx.  3r^_+_2«_^       ^^ 


INTEGRATION   BY  PARTS. 
197.  Integrating  both  members  of 

d{iiv)  =  udv  +  vdii 
and  transposing,  we  have 

/  udv  =  uv  —  I  vdu, (A) 

which  is  the  formula  for  integration  by  parts.  It  reduces  the 
integration  of  udv  to  that  of  vdu,  and  by  its  application  many 
differentials  can  be  reduced  to  one  of  the  elementary  forms. 

EXAMPLES. 
1.  Find    /  x?  log  xdx. 

Assume  u  =  log  x;  then 

dv  =  x^dx,    du  =  — ,    and    v  =   /  x^dx  =  — . 
X  «/  3 


182         DIFFEBENTIAL  AND  INTEGRAL   CALCULUS. 

Substituting  in  (A),  we  have 

/«'  ,               Px^dx       x"  loff  X      X*    ,    ^ 
x"  log  xdx  =  -  log  a;  -J  -^  =  —^ "9  +  ^• 

3.  Find    /sin'^xdx. 

Assume  u  =  sin"^  x;  then  dv  =  dx,  v  =  x,  and  fZw  = 


Vl-i 
Substituting  in  (A),  we  have 


/sin~^  xdx  =  X  sin"^  x  —        -^ =  x sin~^ a:;  +  (1  —  x^^-\- C. 

Find  the  following: 

3.  /  tan-^  X  dx.  x  tan-^  x  —  log  (1  +  a;')*  +  C. 

4.  /  X  cos  X  dx.  X  sin  x  +  cos  x  -\-  C. 

5.  Jxe'^^'dx.  e'^'^f- -,]  +  C. 

pax 

Make     dv  =  e^-'^dx;  .*.  v  =  — ,    it  =  x. 

a 

Sometimes  two  or  more  applications  of  the  formula  are  re- 
quired, as  in  the  next  example. 


'•/ 


x^e°-^dx. 


Vx^      2x       2 ' 
a        ci"  ~^  ft' 


+  (7. 


Make      dv  =  e'^'^dx;     .:  v  —  — ,     u  =  x^,     du  =  Ix  dx, 

a 

y*  „      ,        e°-^x^       „   Pe"'^xdx 
a  da 

!N'ow  apply  the  formula  to  the  last  term,  as  in  Ex.  5,  and  we 
obtain  the  entire  integral. 


GENERAL  DEPENDENT  INTEGRATION.  183 


Va'  -  x' 


xdx 


Make  dv  = •;     .-.  v  =  —  \/d'  —  x\  and  u  =  x\ 

Va'  -  x' 

/'— ^^-  =  -  xWa'  -  x'  +  f'^xia'  -  x'fdx. 

In  a  similar  manner  we  may  integrate  any  binomial  differen- 
tial, or  by  continued  application  of  the  formula  reduce  it  to  a 
simpler  form,  but  by  the  method  of  Art.  211  the  result  may  in 
general  be  obtained  with  less  labor. 

8.  /  x^  cos  X  dx. 

Make     dv  =  cos  xdx;  then  v  =  sin  x,  u  =  x^,  du  =  "ix^dx. 

I  a;'  cos  X  dx  =  x^  sin  a;  —    /  Zx"^  sin  x  dx. 
Again,  make  dv=  -  sin  x  dx;  then  y  =  cos  x,  u  =  3ix'',du  =  Qxdx. 

—    /  Sx^  sin  X  dx  =  Sx"^  cos  a;  —    /  Gx  cos  x  dx. 
Again,  make  dv=—cos  x  dx,  then  v=  —sin  x,  u^=6x,  du-=6dx. 
.'.  ~  /  6x  cos  xdx  ^  —  6a;  sin  x-{-    (  Q  sin  x  dx{=  —  6  cos  x). 

.•.    I  x^  cos  X  dx  =  x^  sin  x-\-3x^  cos  a;— 6a'  sin  x—6  cos  x-{-  G, 

9.  /  log  xdx.  a;(log  a;  —  1)  +  6'. 

/x^  x^ 

x^  log  X  dx.  —  log  x  —  —  -\-  G. 

11.  Jx"  log'  X  dx.  \x^^og^  a:  -  I  log  a;  +  I)  +  C. 


184         DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

12.  fe^x'dx.  e^(x*-4a;='+ 12.^^-24:^+24)  + a 

»"     / L  \    1    (7. 

log  a  V        log  «/ 

-e-^(a;'  +  2a;  +  2)  +  C. 

/  ,       3a;'       6a;        6  \e"''    ,    ^ 
a; h  -^    —  ^  — ■  +  C. 

%  sin-^  X  +  ^!-±^i/r^r^'  +  a 


]3. 

fxa 

""dx. 

14. 

fxh 

>-^dx. 

15. 

/- 

j"^<;/a;. 

16. 

/- 

sin~^  a;  (?a;, 

17. 

■  J  {x 

y  xdx 

+  ir 

a; 


-— -  log  X  -  log  (a;  +  1)  +  a 

18.  ya;'  (log  ti;)'^a;.  "^[(log  a;)'  -  i  log  a;  +  i]  +  a 

19.  /(«'  -  xT^xhlx.  -  |(«'  ~  ''"')*  +  y  '^^~'  a'  +  ^• 

20.  /(«'  +  a;^)-V(^a;.       |  (a'+a;=)*  -  ^log  (a;+  i/;?+^0+^- 

REDUCTION    FORMULAS. 

198.  Eeduction  formulas  are  formulas  by  which  the  integral 
of  a  differential  may  be  made  to  depend  on  the  integral  of  a 
similar,  but  simpler,  differential. 

199.  To  find  the  reduction  formula  for  J  x^  (log  a;)"f?a;, 
■where  n  is  a  positive  integer. 

Assume         dv  —  x^  dx        and       M  =  (logrr)"; 

then  v  =  — —z      and    du  =  ni^ogxY-^ — , 

j-j  + 1  "^    '       X 

Substituting  in  (A),  Art.  197,  we  have 
/a;niog  xYdx  =  ^^^^(y"  _  _^y.^P(log  x^hlx,  (1) 


GENERAL  DEPENDENT  INTEGRATION.  185 

in  which  the  proposed  integral  depends  upon  another  of  the 
same  form,  but  having  the  exponent  of  log  x  less  by  one.  By 
successive  applications  of  this  formula  the  exponent  of  log  x 
is  reduced  to  zero,  and  the  proposed  integral  is  made  to  depend 

upon  the  known  form    /  x^dx. 

Cor.  I.  If  the  given  integral  were  f  X  (log  xYclx,  where  X 
is  any  algebraic  function  of  x,  we  should  have 

fxilogxfdx  =  XX^ogxY  -  JiJ'^ilogxy-'dx, 

where  X^=    I  Xdx. 

EXAMPLES. 
1.     /  x^\og^ xdx. 

Here  ;j  =  3,  ^j  +  1  =  4,  w  =  3  and  n  —  1  —  2.    Substituting 
in  (1),  we  have 

/  .r'  log'  X  dx  =  ix*  log'  X  —  i  J  x'  (log  xydx. 

By  applying  the  formula  to  the  last  term,  etc.,  we  obtain 


/  x^  log'  X  = 


log  a;  -  -  log^  X  +  -^  log  x ^ 


+  C. 


2.  fx'  (log  xf  dx.  ^    (log  xy  -  1  log  a;  +  J3-J  +  C. 

''fWw  ^log.-log(l+.)  +  a 


{i  +  xy  "    '       i-\-x 


186         DIFFERENTIAL  AND  INTEGRAL   CALCULU3. 


200.  To  find  the  reduction  formula  for    /  a^x'^dx,  where 
n  is  a  positive  integer. 


Let 
then 

.-.  Art.  19 


dv  =  a^dx       and       u  =  x"; 


V  =  , and     du  =  nx^~'^dx. 

log  a 


7,      /«- 


x^dx 


ax  n 

log  a       log 

EXAMPLES. 

^x' 


/a^x^~'^dx. 


4 


n^X^dx  X^ 

log  a  L  log  a   '  log'  a      log'  a_ 

=  3,       .•.    /  a^x  dx  =  , /  a^x'dx. 

t/  loff  a       lo2- ««/ 


(1) 


+  C. 


Here  n  —  ^,      . .    ,  ^  .^  ^.^  —  ,  , 

log  rt       log  a^ 

By  further  applications  of  (1)  we  obtain  the  desired  result. 
2.    /  d^x^dx. 


log« 


4.^"  12a;' 


24a; 


24 


log  a       log^  a      log^  a      log"  a  _ 


+  ^. 


/ 


e'^^'x^dx. 


& 
a  L 


r^._3:?i  +  te_4|  +  a 


«  a        a  _ 

201.  To    find   the   reduction   formula   for    /  x^'co&axdx 
and    /  .r"  sin  ax  dx,  where  n  is  a  positive  integer. 

Make  u  —  x"  and     ^y  =  cos  axdx; 

sin  ax 


then 


ti  —  X 
du  =  nx"~hlx     and       v 


/^" 


cos  ax  dx  =  -a;"  sin  ax 
a 


at/ 


~^  sin  ax  dx. 


GENERAL  DEPENDENT  INTEGRATION.  187 

Similarly,  we  fiud 

/x"  sin  axdx  = x^  cos  ax  A —  /  .'c""^  cos  ax  dx. 
a  ad 

Hence,  in  either  case,  the  integral  can  be  made  to  depend  on 
the  known  form    /  cos  ax  dx  or    /  sin  ax  dx. 

EXAMPLES. 

1.  /  X'  cos  X  dx.       x^  sin  x  +  3a;^  cos  x  —  6»  sin  x  —  ^  cos  x  -{-  C. 

2.  /  x""  sin  X  dx. 

—  X*  cos  X  +  4:X^  sin  x  -\-  12.t'  cos  x  —  24:X  sin  x  —  24  cos  x  -{-  C. 

202.  To   find   the   reduction    formula    for    ^sm~'^xdx, 
^Ttan^^  a-  dx,  etc.,  where  X  is  an  algebraic  function  of  x. 

Make       u  =  sin~^  x         and     dv  =  JTdx; 


then  du  =  -^  "^ — -_      and      v  =    I  Xdx  =  X^  (say). 


t  X 
Vl—X' 

X,dx 


I  Xsin  ^  xdx  =  X^  sin~^  x  —   I  ■ 


VI  ~ 


1. 


EXAMPLES, 
a:^  tan~^  x       x^      log  (1  -f  x^^ 


r  1.      -X      1                      ^  tan-^  X       x^      log  (1  -fa;')    ,    „ 
y  a;'  tan  ^  a;  rZa;.  ^ —  +     ^  ^  g- -'  -f  (7. 


a; 
Here  X=x^    and     X,  —  — . 


2-  ft^^lp:^,      ^  tan-i  a;-  i(tan-i  xf  -  ^  log  (1  +  a;')  +  C. 

3.  J  x""  seo-'^  X  dx.  |a;'sec~^a;— i-(a;'— l)^a;— i-log  (a.'+V'a;'  — 1-f  C 

Eeduction  formulas  for  binomial  differentials  are  deduced  in 
Art.  215. 


188         DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

203.  To  integrate        ; -. 

a  +  0  cos  X 

dx  dx 


^  a[cos'  -  +  sm^  -  j  +  &(^cos^  -  -  sm''  -  j 


dx 


{a  +  V)  cos'  *o  +  («  —  ^)  sin'  - 
sec''  ~dx 


<?a; 


(a  +  5)  +  («  -  J)  tan'  - 
[tan 


.Z^      ^ 


=  2 


''-Ja  +  boo,x      "     /       («  +  5)  +  («_j)tan'|' 

which  is  readily  reduced  to  the  form 

/dv                  /*    dv 
^— — 5     or      /  -i S-,    according  as  a  >  or  <b, 
r  -\-  V             t/  r  —  v 

/dx 
— —.—-. can  be  found. 
a-\-o  sm  X 


m.^   M         rr,  .L.  ^  dx  ^  dx 

204.  To  integrate  -. and    . 

sm  a;  cos  x 

r  dx  _  n      \dx       _  r 

J  sin  a;  ~  J  sin  \x  cos  \x     «/       tan  \x 
1 4^  =  log  tan  ix  =  log  y  - 


ScC       o*C'    (yCtju 


—  cos  a? 


sin  a;  *=  ^  '    1  +  cos  a; 


r  dx    _     r         dx 
g^^^'  J  ^«^^"  7  sin  (I -a;) 

=  -  log  tan  (-f  -  I;  4-  C^. 


GENERAL  DEPENDENT  INTEGRATION  189 


APPKOXIMATE  INTEGEATION. 

205.  The  number  of  differentials  which  can  be  integrated 
xactly  is  comparatively  very  small,  yet  the  approximate  value 
of  the  integral  of  any  differential  may  be  found  when  the  dif- 
ferential can  be  developed  into  a  convergent  infinite  series  each 
of  whose  terms  is  integrable.  This  is  the  last  resort  in  separat- 
ing a  differential  into  its  integrable  parts. 

EXAMPLES. 
Find  the  approximate  integral  of  the  following: 
^     ,  dx  X        x^        x'        X*    ,     ^ 

Expanding  — -—  by  division,  we  have 

1  1        X    ,   x"^      x^   ,     , 

+  ::?-  T4  +  etc.; 


a  -{-  X      a       a"^      a        a* 


y  =    /( ;-f  —  — T  +  etc.  Jc?a;. 


o     7  ^^  x\   x"      x'    ,      . 

-^^^  =  1+1^-  ^  =  ^-3-  +  ^-y  +  ^*^- 

By  division, :; — ; — 5  =  1  —  x^  4-  x'^  —  x''  -\-  etc. 

•^  1  -\-  X  ' 

^     ,  dx  x'      ,       Sx'  3  .  5:r'       ,     ^ 

^-'^^^'vfT^'    ^  =  ^-2T3+27475-3:4:677  +  ^*'- 

By  the  Binomial  Theorem, 

—  l——--\-  - — -  —  etc. 


Vl  +  x'  2    '   2.4 

4.  di/  =  x\l  —  x'^Ydx.  y  =  ^x^  —  ^x^  —  ^x^  —  etc. 

Develop  (1  —  a;')*,  multiply  by  o^dx,  etc. 


190         DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

5.  dy  =  a;'(cos  x)dx.  ^"T"  12  "^193"  ®^°" 

6.  dy  =  X'  sin-^  xdx.  y  =  x'L  +  —  +  ^  -f  etc, 

206.  Development  of  Functions  by  Exact  and  Approxi- 
mate Integration.  Two  or  three  examples  will  suffice  to  illus- 
trate  the  process. 

EXAMPLES. 

— - —  =  log  {a  +  x),  exactly  (omitting  C); 

CI  ~\-  u^ 


r    dx         X        x^    ,    X  ,  •      i.  ^ 

I  = -„  +  -—i  —  etc..  approximately. 


X  X  X 

'.  log  (a  4-  x)  = 7r-o-\-  ^—  e^c. 


/  -  ^=  =  log  (x  -{-  Vl  -\-  x"),  exactly; 
^    Vl  +  x' 

r      dx                    x"    .    ^x''         ,  -4-1 

/  — =  X — etc.,  approximately. 

t/    Vi  _|_  a;2  6        40 

.-.  log  {x  +  vr+^)  =  ^ - y  +  J-  -  etc. 

/ '- — -  =  tan"^  X,  exactly; 

/»     dx                    x^       x"        x'  .         , 

/  r+^  ~^"~y'^"5 f"^        '  approximately. 


CHAPTEE  IX. 
INTEGRATION  —  {Continued). 

INDEPENDENT   INTEGRATION. 

207.  Increments  Deduced  from  Differentials.  We  have 
seen  that  the  increment  of  a  function  is  the  sum  of  the  differ- 
ential and  the  acceleration;  hence,  when  tlie  former  is  known^ 
we  can  find  the  differential  by  simply  removing  the  acceleration. 
Taylor's  formula  enables  iis  to  reverse  this  operation  in  many 
cases,  and  find  the  increment  when  the  differential  is  known. 

Let  u  =  f[x). 

Increasing  x  by  //,  we  have,  by  Taylor's  formula, 
u  +  Jit  =  f{x  +  7i) 


Au^f{x  +  h)-f{x) 


=  /'(^)'^+/'>/^+  .  .  ./n^)^.  .  .,       (A) 


in  which  du  =  f'{x)h. 

Therefore,  when  the  differential  of  a  function  of  x  is  known, 
the  increment  maybe  found  by  taking  the  successive  derivatives 
of  the  differential  coefficient,  and  substituting  them  in  (A). 

When  f^{x)  =  0,  and  each  of  the  subsequent  derivatives  of 
f(^x)  =  0,  the  series  will  be  finite  and  express  the  exact  value  of 
Jii;  otherwise  the  series  will  be  infinite,  and,  if  convergent,  will 
give  the  approximate  value  of  /}u. 

191 


192        DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

EXAMPLES. 

1.  If  du  =  (a;*  —  5x  +  6)dx,  what  is  the  value  of  ^u? 
Here  f\x)  =  x"^  —  5x  -\-  6.     Differentiating  this^  we  obtain 

f'\x)  =  2x  -  5,  f'"{x)  =  2,  p^{x)  =  0.      Substituting    these 
values  in  (A),  we  have 

^u  =  {x'  -5X  +  6)h  +  {2x  -  o)~  +  ^. 

2.  If  du  =  {x^  —  Sx  -  10)h,  what  is  the  value  of  Jw  ? 

^u  =  {x'  -  Sx  —  10)h  +  {2x  -  3)|-'  +  ^. 

3.  If  du  =  («'  —  7x^  +  12;i:)(?.T,  what  is  the  value  of  au  ? 

au  =  (Sx'  -  Ux  +  13)^^+  (3a;  -  7)|^  +  ^*  • 

/i  3  4; 

4.  Find  aw  when  du  =  sin  a;  (?a;. 

h'         .       ¥  h* 

au  =  cos  X sm  a;  — -  —  cos  a*  -—  +  etc. 

2  6  34 

5.  If  du  =  (V'l  -{-xjdx,  by  how  much  will  w  be  increased 
when  .T  is  increased  by  h  ? 

Z/w  =  (4/1  +  a;)A  +  (1  +  a;)~*  —  -  (1  +  x)'^  —  +  etc. 

6.  Find  au  when  6??^  =  log  x  dx. 

""^  ~  2x        Qx'  +  i2a;^       20a;^  +  ®  ""• 

7.  A  function  is  increasing  at  the  rate  of  4:X^dx;  find  its  suc- 
ceeding increment.  Ju  =  4x^71  +  Sx'^h^  -\-  4x¥  -\-  h\ 

8.  At  the  end  of  t  seconds  the  velocity  of  a  body  is 

-1  =  (3f  -  2t)  ft.  per  second; 

Cti/ 

find  the  distance  it  will  travel  the  following  second,  dt  being  the 
unit  of  time.  ^s  =  (3f  -  2t)dt  +  {dt  -  l)df  +  dt\ 

9.  The  rate  of  acceleration  of  the  velocity  of  a  body  is 

dv 

— -  =  (3/  -f-  4)  ft.  per  secdnd; 


INTEGBATION.  193 

find  the  increment  (1)  of  the  velocity  [v),  and  (2)  of  the  distance 

{s)  for  the  following  second. 

Av  =  {dt  +  4)dt  +  ^dt\      V  =  f{?>t  +  4)^/^;  =  |f  -\-U-\-G, 

As  ^  (|f  +  4if  +  C)di  +  {\t  +  2)f7i!'  +  \dt\ 

208.  Increments  as  Definite  Integrals.  In  Fig.  5,  where 
u  =  area  of  OBPA,  An  =  the  area  of  BCP'P,  which  is  evi- 
dently the  integral  of  du  between  the  limits  x  and  x  -\-  Ti.  In 
general 

f\x)dx=f{x  +  h)-f{x). 
For,  since     df{x)  =f'{x)dx,    ff'{x)dx  =f{x)  +  0. 

/^x+h  |_  _  x+h 

/  /'(^)^a'  =    fix)  +  Cj     -/(a;  +  h)  -  f{x). 
*y  X  X 

Therefore  (A)  may  be  written 

/  f'{x)dx  =r{x)h+r\x)-^+f-'{x)^ . .  .rixh-. .  .(b) 

By  this  formula  we  can  obtain  exactly,  or  in  the  form  of  an 
infinite  series,  the  definite  integral  of  any  function  of  a  single 
variable,  and  the  operation  does  not  involve  the  reversing  of  any 
of  the  formulas  for  differentiating.  But,  in  general,  this  method 
is  much  inferior  to  that  of  dependent  integration,  since  by  the 
latter  many  differentials  can  be  integrated  in  finite  terms  which 
by  the  former  could  be  expressed  only  in  the  form  of  an  infinite 
series.  However,  it  forms  an  important  part  of  the  theory  of 
differentials  and  integrals,  and  is  often  useful  as  a  method  of 
approximation. 

More  convenient  formulas  for  practical  purj)oses  will  be 
derived  from  (A),  but  before  doing  so  let  us  apply  (B)  to  the 
following  illustrative  examples. 


194         DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

1.  Find  the  area  BCP'P,  the  equation  of  APP'  being 
A 


y  =  x'  -lx-\-  12,  where  x=  OB,y  =  BP,  smd  BC=h  or  dx. 
Let  u  =  area  of  OB  PA,  then  du  =  ydx. 

y=f'(^x)=x'-7x  +  12,    f"{x)  =  2x-l, 

f'"{x)  =  2,    p^{x)  =  0. 

Substituting  in  (B),  we  have 

j  {y)dx  =  [x'  -  '7x  +  12)h  +  {2x  -7)y  +  y  =  ^^^^  °^  BCP'P. 

Cor.  I.  To  find  the  area  of  OB  A  we  make  x  =  0  and  h  = 
OE  =  3,  and  obtain  13^.  To  find  the  area  of  EnF,  we  make 
X  =  OE  =  2,  wadi  li  =  EF  =  1,  and  get  —  f 

Let  the  student  solve  the  following  in  a  similar  manner. 

2.  The  equation  of  a  curve  is  ?/  =  x"^  —  %x'  -\-llx  —  %',  find 
the  areas  of  the  two  sections  enclosed  by  the  curve  and  the  axis 
of  X.  i;  —  i. 

209.  A  More   Convenient   Series.      In    (A),   by   making 

f'{x)dx  for  f{x), 
we  have 
rf'{x)dx  =/(0)  +/'(0).r  +/"(0)^+  .  .  ./"(O)^.  .  .      (C) 

This  formula  may  be  obtained  by  developing  f'{x^  by 
Maclaurin's  formula,  multiplying  by  dx,  and  integrating  each 


INTEGRATION.  195 

term  separately,  but  as  we  are  now  exemplifying  the  method  of 
independent  integration,  we  will  apply  (C)  directly  to  one  or 
two  examples. 

1.  Find    Asa;'  -  14a;  +  b)dx. 

Here  f'{x)  =  Sx^  -  Ux  +  5,    .-.    /'(O)  =  5; 

f'\x)  =  6x-  14,  /.   /"(O)  =  -  14; 

f"'{x)  =  6,  .:f''\0)  =  Q. 

Substituting  in  (C),  we  have 

y  (3a;'  -  Ux  +  5)dx  =  x'  -  7x'  +  5x. 

2.  Pix'  -  Qx"  +  l)dx.  ix'  -  2x'  +  7x. 

3.  PiSx*  -  2x'  +  x')dx.  p'  -  ^x'  +  ix\ 

210.  Bernouilli's    Series.      In   formula   (B),   by   making 
f'{x)dx  =  —    I  f'{x)dx,  we  have 

£f\x)dx=f{x)X  -r{x)^  +..._(_  lYr{^)y  .  .(D) 

This  formula,  called  Bernouilli's  Series,  like  formulas  (B) 
and  (C),  shows  the  possibility  of  expressing  the  integral  of 
every  function  of  a  single  variable,  in  terms  of  that  variable, 
since  the  successive  derivatives  f"{x),  f"'{x),  etc.,  can  always 
be  deduced  from  f'{x).  Hence,  in  all  cases  where  the  series 
are  finite  or  infinite  and  convergent  the  integral  may  be  ob- 
tained exactly  or  approximately. 

In  finding  Jf'{x)dx  by  (B),  (0),  or  (D)  the  limits  of  the 

difference  between  the  approximate  value  fou.nd  and  true  value 
may  be  determined  as  in  Ag  of  the  Appendix. 


196         DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 


INTEGRATION  BY  INDETERMINATE   COEFFICIENTS. 

211.  The  process  of  integrating  binomial  and  trigonometric 
differentials  by  successive  reductions  is  generally  very  tedious, 
and  it  is  our  purpose  now  to  present  a  method  which  is  gener- 
ally less  laborious,  and  which  is  also  applicable  to  many  other 
classes  of  differentials. 

Let  u^vdx  be  the  differential  to  be  integrated,  where  u  and  v 
are  functions  of  x,  and  let  us  assume 

fiC'vdx  =  W+Y{x)  +  k  Cn,^v^dx,  .     .     .     .     (E) 

in  which  it  is  required  to  find  the  function  f{x)  and  the  con- 
stant h. 

In   examples  where     /  iCvdx  can  be  expressed  under  the 

form  of  w''+y(a;),  we  shall  find  ife  =  0;  and  when  this  is  not  the 

case,   /  u'^vdx  will  be  determined  in  terms  of    /  u^v^dx,  which 

we  can  generally  make  of  a  more  elementary  character  by  as- 
signing suitable  values  to  s  and  v^. 

Another  advantage  of  expressing  the  required  integral  under 
the  form  of  (E)  arises  from  the  fact  that  «''+y(a;)  often  vanishes 
for  the  desired  limits  of  integration,  in  which  case  the  definite 
integral  depends  on  the  last  term  only. 

Differentiating  (E)  and  dividing  by  u^dx,  we  have 

v  =  {r^\f-gf(x)-^nf'{x)^Tcv,,e-\.     .     .     (F) 

The  simplest  and  easiest  method  of  solving  this  equation  for 
f{x)  and  Tc  is  by  indeterminate  coefficients,  as  illustrated  in  the 
following  examples. 

212.  Case  I.  When  h  =  0; — Independent  Integration. 


EXAMPLES. 


1.  Find    /"(I  +  x^y'^x'-dx. 


INTEaRATION.  197 

Comparing  this  with  (E)  we  have  u  =  1  ~\-  x',  r  =  —  ^^ 
—  =  %x,  and  V  =  a;^     Substituting  in  (F),  we  have 

x^  =  xf{x)  +  il-{-xY'{^)  +  ^v,{l  +  xT\      .     .     (1) 

The  first  member  of  this  equation  being  of  the  fifth  degree 
the  second  must  be  also;  hence, /(.r)  must  be  of  the  fourth  de- 
gree, and  since  u  involves  only  the  second  power  of  x,  we  may 
assume 

f{x)  =  Ax*  +  Bx'  +  (7,     .-.  f{x)  =  4: Ax'  +  2Bx. 

Substituting  in  (1),  and  arranging  in  reference  to  x,  we 
have 

x'  =  5  Ax'  +  (3^  +  4:A)x'  +  (C  +  2B)x  +  kv,{l  +  x'Y^K 
1  =  5A,    3B  =  -  4.A,     G=  -2B,     k  =  0, 
or  A  =  \,        B  =  -^,      C=^%. 

These  values  determine  f{x),  which  substituted  in  (E)  gives 

f{i  +  xY'^\ix  =  (1  +  x^yi^x*  -  ^sx'  +  ^x]  +  a 

2.  Find  /.-(I  +  x^)^dx.  -  (1  +  .f(^.  -  xli)  +  a 

fill 

Here  ic  =  1  -^-x",  r  =  i,  —  "=  ^^>  ^^^  ^  —  ^~^'^  ■'•  (^)  gives 

x-'=Zxf{x)  +  {l^xY'{x)  +  ]cvX\  +  x-f-^.    .     .     (1) 

In  order  that  the  two  members  may  be  of  the  same  degree 
we  assume 

t\x)  =  Ax-^+  Bx-'+  Cx-';     .'.  f'{x)  =  -  hAx'^-^Bx-'-  Cx-\ 

Substituting  in  (1),  and  arranging  in  reference  to  x,  we  find 
A  ^  —  ^,  B  ^=  -^j,  C  =  0,  ^•  =  0;  this  determines  f{x);  which 
substituted  in  (E)  gives  the  desired  result. 


198        DIFFERENTIAL  AND  INTEGBAL  CALCULUS. 

A  careful  inspection  of  the  previous  examples  suggests  the 
following  rule  for  determining  the  form  of  f{x)  for  binomial 
differentials  of  the  form  {a  -\-  bx^yx^dx,  v  being  a;'": 

I.  When  m  is  positive  the  highest  exponent  of  x  inf{x)  will 
be  m  —  7^  +  1. 

II.  When  m  is  negative  the  algebraically  lowest  exponent  of 
X  in  fix)  will  be  —  m  -\-  1. 

III.  The  remaining  exponents  decrease  or  increase  alge- 
braically by  n. 

The  rule  is  also  applicable  when  ?«  is  a  polynomial  in  which 
n  is  the  highest  exponent  of  x,  provided  that  the  exponents  of  x 
in  f{x)  increase  or  decrease  by  the  least  difference  between  the 
exponents  of  x  in  u. 

When  r  is  a  fraction  and  u  a  polynomial  of  a  higher  degree 
than  the  second^  the  differential  cannot  ordinarily  be  inte- 
grated; or,  more  accurately,  its  integral  cannot  ordinarily  be 
finitely  expressed  in  terms  of  the  functions  with  which  we  are 
familiar.  The  exceptional  or  integrable  cases  are,  in  general, 
where  u,  v,  and  r  are  such  that  it  is  possible  for /(a;)  to  have  as 
many  coefficients  as  there  will  be  independent  equations  between 

the  coefficients  in  equation  (F),  and  where  k  j  v^uHx  is  0  or 

one  of  the  integrable  forms.  In  a  differential  of  any  given  form 
the  conditions  of  integrability  may  often  be  determined  by  the 
present  method. 


■  J  a;*  ^x'  +  6a;  +  15*  -45  \x'      2x'       6x1 

Here  u  =  x"^  -{-  6x  +  15,  r  =  —  |-,  and  v  =  x"^;  hence  the 
lowest  exponent  of  x  in  f{x)  will  be  —  4  +  1  =  —  3,  and  the 
others  will  increase  by  1,  giving /a;  =  Ax'^  +  ^^'^  +  C^'^  +  -^« 

The  process  can  also  be  applied  to  many  differentials  in 
which  V  is  a  polynomial,  as  in  the  next  example. 


/ 


3a;'  +  5a;  4-  5  ^  . fSx  4-  1  ^ 


INTEQBATION.  199 

Here  u  =  x^  -\-  2x  -\-  %,  r  =  —  ^,  v  =  'ix^  +  5a?  +  5,  and  f{x) 
is  of  the  form  Ax  -\-  B. 


■    r    dx  X     rs  4  ,  4  ,  ,  ^ 


(1  +  xy  (1  +  xy  LIS  3 


+  C'. 


Here  u  =  1  -\-  x'',  r  =  —  |^,  v  =  1;  and  making  v^  =  1,  s  = 
-  i,  (F)  gives 

1  =  -  bxfix)  +  (1  +  a.-'0/'(:i-)  +  ^'(1  +  xy, 

where /(a;)  is  evidently  of  the  form  Ax^  -\-  Bx^  +  Cx. 

213.  We  have  seen  (Art.  185)  that  sin*" a;  cos"- a;  ffe  can  be 
easily  reduced  to  an  integrable  form  when  either  m  or  n,  or 
both,  are  positive  odd  integers,  or  when  m  +  n  is  an  even  integer 
and  negative.  In  such  cases,  m  and  n  being  integers,  the  inte- 
gration may  be  effected  by  the  independent  method,  as  in  the 
two  following  examples;  but  this  method  of  integrating  such 
differentials  is  introduced  and  recommended  chiefly  for  its  bear- 
ing on  the  cases  in  which  the  above  conditions  do  not  exist,  and 
which  are  usually  solved  by  succes^sive  reductions. 


^ 


sin^  X  cos*  X  dx.       —  cos^  x 


'sin*  X       4  sin'^  x        8  ' 
9      "^""63        ^3l5 


+  C. 


We  may  make  u  =  cos  x,  r  =  4o;    then  ^7  =  —  sin  a;  and 
V  =  sin^a;.     Substituting  in  (F),  we  have 

sin^  X  =  —  5  sin  xf{x)  -\-  cos  xf'{x)  +  Jcv^  cos*"*  a;, 

where  f{x)  is  evidently  of  the  form  A  sin*  x  4-  B  sin''  x  -{-  C, 
and  hence /'(a;)  =  {4: A  sin'  x  -f-  2B  sin  x)  cos  x. 

.%  sin^  a;  =  —  5^  sin^  x  —  5B  sin'  x  —  bC sin  x 

-j-  {4:A  sin'  X  -\-  2B  sin  x)  cos^  x  +  etc. 

Now,  substituting  1  —  sin''  x  for  cos^  x,  reducing,  and  arrang- 
ing with  respect  to  sin  x,  we  have 


200        DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 


(1  +  9.-1)  sin^  X  +  (75  -  4^)  sin'  x^{bC-2B)  sin  x 

-\-  hv^  cos^~*a;  =  0. 


^  =  -1,     B=-^S>     C=--,\^,    k  =  0. 


dx 
cos"  X ' 


cos  a; 


--  sin*  X  —  -  sin'  x-\-l 
15  o 


du 


+  a 


Make    w  =  cos  x,    r  =  —  6,     ~  =  —  sin  a;,     y  =  1,    s  =  0, 
^7,  =  1,  and  we  have  from  (F) 

1  =  5  sin  xf{x)  -\-  cos  xf'{x)  +  ^  cos^  a;. 
Assume  f{x)  =  A  sin'  x -\-  B  sin"  a;  +  C'sin  x. 

.'.    f'{x)  =  bA  sin*  X  cos  a;  +  35  sin'  x  cos  a;  +  (7  cos  x. 
Substitute,  reduce,  etc.,  and  we  find 

A  =  ~„     B=-^,     C=l,    k  =  0. 

lO  o 

This  method  of  integration  can  often  be  applied  to  other 
classes  of  differentials,  as  in  the  next  two  examples. 


J'x'  log' 


xdx. 


|(log'^-  glog^  +  ^J  +  C. 


25 


Make  n  =  x,  r  =  4,  v  =  log°  x,  and  assume 
f{x)  =  A  log'  x  +  Blogx  +  a 


9. 


/ 


e'^'^x^dx. 


3x'    .   6x 


„axri ^  +  —  _—    +a 

a        a         a        a   ' 


du 


Make  «  =  e"^  r  =  0,  5  =  1 ;  then  v  =  e"^a;',  and  -^  =  ae"""; 

dx 

substituting  in  (F)  and  dividing  by  e"^,  we  have 

:c'=af{x)^f'{x)^lcv^, 
where  we  evidently  have/(.r)  =  Ax^  +  J^^^  +  ^'^  +-^- 


INTEGRATION.  201 

Find : 
/''    x^  clx  , (\  2\ 

n    x^dx  , (I  14  1.2.4\ 

«^    Vl-x^ 

ri      -^V?^  +5.7^  +3.5.7'^  +1.3.5.  7^  +  ^* 
13.  y  a;-2(l  +  xT^^dx. -^ (-  +  2-;^  +  C. 

Assume  A^)  — 1~  ^^• 

J    ^a  +  hx'  V  b      '    ¥  llbb 

•^    («  +  l)xy  {a  +  Ja;^)A  da"  ^  aJ  ^ 

17.  y  x\2  +  3a;^) Va:.  ^-^^ (^_  «;«  _  _  ^.  +  __  j  +  (7. 

18.  y  a;-*(l  -  2x')-^dx.         -  (1  -  2a;=)*(-^±^')  +  (7. 

«-,       r{2x'-5x*-ex-\-7)dx 

''■  J     (.'-5.T^'     ■      (^'  -  ^^  +  2)-(-  -  ^)  +  ^- 

22.    /  sin^  X  cos'  a;  (^a:.  -J  sin'  a;(cos^  a:  +  -J)  +  C 

Make  w  =  sin  a:,    r  =  5,     v  =  cos^  a;. 


202        DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

1 


/'sin  a;  , 
23.     /  ——dx. 


24. 


cos  a; 


/  x"  log' 


3  cos  X 


sin*  x-\-  ^  siii^  X  —  % 


xdx. 


6 


log'  X--  log^  a;  +  -log  a;  -  — 


25.     /  e'^^'x^dx.     e" 
f^dx. 


'x'       ix'      4.3a;'  _  4 .  3 .  2a;      4.3.2.1' 
«        «'  «'  «*  a* 


26 


1^ 

2? 


log^  a;  +  log  a;  +  - 


214.  Case  II.  When  k  is  not  =  0; — Dependent  Integra- 
tion. 

EXAMPLES. 

1.  /a;Xl  -  ^I'dx.       (1-^f  [1^  -  ^  -  ^]+  ^sin-a;  +  (7. 

The  differential  a;*(l  —  x'^ydx  has  the  same  binomial  factor 
as  (1  —  x^)~^dx,  whose  integral  is  sin~^a;;  hence,  by  expressing 
the  former  in  terms  of  the  latter,  the  required  integral  may  be 

obtained  in  terms  of    /  (1  —  x^)~^dx.     Thus, 


x'{l  -  xf  =  x\l  -  a;^)(l  -  x')-^  =  {x*-x'){l-x')-K 

*.*.  u  =1  —  x^ ,  r  =  —  i,  3—  =  —  2x,  V  =  X*  —  x%  which  sub- 

stituted  in  (F)  gives  (making  s  =  r  and  v^  =  1) 

x^  -x'  =  -  xf{x)  +  (1  -  xY'i^)  +  k, 

where  f{x)  —  Ax"  +  ^^*  +  C'a;;    and  proceeding  as  before,  we 
find  J  =  I,  5  =  -  ^V^  ^  =  -  T*6>  and  k 


1 

16' 


/ 


6?a; 


(1  +  =^^)' 


(1  +  xy 


+  fa;]  +  f  tan-i  a;  +  (7. 


^w 


Here  v  =  1  +  a;',  r  =  —  3,  -=-;-  =  2a;>  «;  =  1,  and  in  order  to 


INTEGRATION.  203 

/dx  r    dx 

— — — ^TT  in  terms  of   /  :; — ; — ;[=  tan"^  x'\  we  make 
(1  +  x)  ^    1  -\-  x^  -^ 

V ^=  1 ,  s  —  —  1,  s  —  r  —  'Z,  and  (F)  becomes 

1  =  _  4:r/(,^')  +  (1  +  x'^)f\x)  +  h{l  +  3a;^  +  x'), 
where /(a;)  —  Ax'  -{-  Bx. 


In   order  to   express    the  integral  in  terms  of    /  - 


Vl  — 


=  loff 


we 


make  u~^  =  xVl  —  x''  =  Vx"  —  a;\ 


5  =  —  i,  then   u  =  x'  —  x\   r  =  —  h   -t~  =2x  —  4a;'    v  —  —, 

ax  X 

and  (F)  becomes 

a;-  =  (^  _  2a;V(^)  +  {x'  -  xY'{x)  +  k, 
where /(ic)  =  Ja;"^  +  Bx~^  +  C'a;"*,  since  a;^  is  a  factor  of  m. 


a;'  +  8a;  +  21 


J  (x'  - 


{x'  -  4a;  +  9) 
Here 


dx. 


flit 

It  =  x'  -  4:x  +  9,  r  =  -  2,  —  =  2x  -  4,  and  v  =  x' -\-8x +  21; 

dx 

hence  we  have  from  (F),  making  s  =  —  1, 

x'  +.8a;  +  21  =  (4  -  2a;)/(a;)  +  {x'  -  4a;  +  9)f{x) 

+  kv^{x'  -  4a;  +  9).     .     (1) 

The  second  member  of  this  equation  must  be  of  the  third 
degree;  but  if  we  make  v^  =  1,  the  solution  will  be  impossible, 
let  us  therefore  assume  that  k  =  1  and  v^  =  Cx  -{-  D;  we  may 
then  make /(a;)  =  Ax -{-  B  and  f'{x)  =  A.     Substituting  in  (1), 


204         DIFFERENTIAL  AND  INTEOBAL   CALCULUS. 

we  find  .4=1,  ^  =  -  V,  C  =  1,  and  Z*  =  V-     Hence  the  re- 
quired integral  is 

?,{x  -  7) r{x+J£jdx^ 

2(.ic'  —  4a;  +  9)  "^  i/  a;''  —  4a;  +  9 

_       3(a;  -  7)  r  {x  -  2)dx  P ^J^dx 

—  3(a;2  -T/^x  +  9)  "^  t/  a;'  —  4a;  +  9  "^  t/  a;'  —  4a;  +  9 

=  -; — ^ — : — —^  +  i  lo^  (a;''  — 4a;  +  9)  H — -—  tan  M  — ~    +  C. 

The  solutions  of  the  three  following  examples  illustrate  the 
manner  of  integrating  sin™  x  cos"  x  dx  when  the  conditions 
stated  in  Art.  213  do  not  exist. 

5.    /   ■  ^    .  .   ,     [f  cos'  a;  —  I  cos  a-]  +  f  log  tan  ^  x -\-  C. 

J  sm   x  snr  x^ 

^        ;1  ^  -,    dU 

Here  we  make  ?f  =  sm  a;,  r  =  —  5;  then  v=l  and  -r-  =  cos  x. 
Making  v,  =  1  and  5  =  —  1,  (F)  gives 

1  =  —  4  cos  a/(a;)  +  sin  xf'{x)  -\-  Jc  sin*  x. 

Making  sin*  x  =  {I  —  cos''  xY,  f{x)  =  ^  cos'  a;  -f  ^  cos  x, 
and  proceeding  as  in  examples  6  and  7,  Art.  213,  we  find  A  =  ^, 
B=  —%,  and  h=^%. 

/-•_^^^      1_      ^^^3^_5cosa;)+f   f4^.    (Art.  204) 

J  sm'  a;       sm*  x^^  ^    '   '^  t/  sm  a;     ^  ^ 

This  example  may  also  be  solved  like  the  following  one. 


f*   dx 

«/    COS    X 


Make  w  =  sin  »,  r  =  0,  then  -5—  =  cos  x,  and  v  =  cos*''  x. 

dx 

.'.    cos"^  a;  =  cos  ^/(a;)  +  sin  xf'{x)  -f  ^Wj  sin*  a;,     .     (1) 


INTEQBATION.  205 

where         f{x)  =  A  cos'®  z  -\-  B  cos'^  x  -\-  C  cos~^  a; 

and  /'(^)  =  (6^  cos"^  x-\-4:B  cos"^  x  +  2C  cos"^  x)  sin  2;. 

Substituting  in  (1),  making  sin'  x  =  1  —  cos'  x,  reducing, 
etc.,  we  have 

6A-1      4:B  -  5A      2(7-35         C 

; —  + 1 + 5 h  A;v,  sm*  X  =0. 

cos  X  cos  X  cos  a;         cos  x  * 

.*.  A  =  \,  B  =  -i^,  C—  i|,  and  (making  s=0,  v^  = )  k  =  C, 

\  COS  it// 

dx  sin  X         5.1  sin  a; 


r_dx__ 
J  cos'  a;  ~  .  .„  

J.   r  dx 

2  «/  cos  X 


cos'  a;      6  cos"  a;      6  . 4  cos*  a; 

.    5 .  3  . 1  sin  a;      5.3.1 


6.4.2  cos'  X      6 .  4  .  2  «/  cos  a; 
To  integrate  the  last  term  see  Art.  204,  page  188. 

7.     /  sin*  X  cos*  x  dx. 

r     cos"  a;  ,  cos' a;      3  cos  a;"]        3  ,  ,  x      ^ 

l-~8"+~16"  +-64-J-j28(sm  a;  cos  a;-a;)  + (7. 


sm^a; 


Make  u  =  sin  x,  r  =  2,  then  v  =  sin' a;  cos*  a;  =  cos*  a;— cos^a;; 
also  make  5  =  2  and  i\  =  1,  then 

cos*  X  —  cos°  a;  =  3  cos  xf{x)  +  sin  xf'{x)  -\-lc.  .     .     (1) 

Make /(a;)  =  A  cos'  x  -\-  B  cos'  x-\-  C  cos  x,  find  f'{x),  sub- 
stitute in  (1),  for  sin'  x  write  1  —  cos'  x,  etc.,  and  we  find 
A  =  —hB=^^,(J=  -i^,  and  k  =  /j. 

/,      ,          .   3     r     cos'  a;   ,   cos'  a;   ,   3  cos  x~\ 
sm*  X  cos*  xdx  =  sm  a; —  -4 — — — 
L         8              16             64     J 

To  integrate  the  last  term,  see  ex.  11,  Art.  185. 


206         DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 
I        x'^dx 


9. 

10. 
11. 


dx 


[- 


¥"'-=''  + \  ^--' I 


Jo 


r'Va'  -  x' 


Va'  -x"  ,     1 


log 


_       log(-l) 

2a'      ' 


2aV      '   2a'^"^  a+Va'-x'A_, 
fVaF+x' dx.     ^Va-"  +  x'  +  ^^—  log  {x  +  Va"  +  a:^  +  C 


3a* 


-{2x'  +  5a^)  V«^  +  «'  +  -^  log  {x  +  ^a:"  +  a^  +  C. 
8  o 

12.  J\a-  -  x^fdx.        |(5fl^  -  2x')V^^^''  +  ~  sin-i  |  +  G. 
13. 


a;''^:?^; 


V2aa;  —  x" 


[- 


|/2aa;  —  a;' 


a;  +  3a\      3a^ 


2 


J  + 


vers" 


1;^-^ 


=  %7Ta\ 


14 
15 

16 


.     /     x^V?.ax  —  x"  dx. 
«/o 

.     /     x^V^ax  —  x"  dx, 

4/0 


^dx 


17. 


Vl  —  a 


fTra. 
^Tta''. 

2.4\2 


1.3.5/7r\ 
2.4.6X2/' 


Jo    ^1-^' 

18.  /  1^3  +  2a;  +  x""  dx. 

4/3  _f.  2a;  +  a;{^4^)  +  log  (2a;  +  2  +  24/3  +  2a;  +  a;0  +  C. 

19.  y  VlO  +  3.r  -  .r"  rfa;. 

,,^    ,    ^ 5/^23;  —  3\       49    .     ,  2a;  -  3    ,    „ 

|/10  +  3x  -  ^\— ^— j  +'"8-  sm-^  -^ h  a 


INTEGRATION.  207 

20.      /  ^dx.  -W-TT. 

91      C    ■  ^^^  ^ u  —  \    -i- J.  n 

~^'  J  (a'  _[_  x'V  2aHa'  4-  x'^   "*"  2a'  ^^     a;  +  ^* 


-  / 


x"  —  .t'  +  21  , 
(a;-  +  3) 

fi7;3  _|_  1  Orr'^  4-  22a;  4-27        1 


+  -^taii-^-^+a 

24/3         1/3 


23. 


/^    (3a;  +  2)dx 
J   {7?  -  dx  +  3)'^' 


13a:  -  24         ,26         _j  2a;  -  3 
3(a;^-3a;  +  3)  +3  1^3  "vT  +  ^' 

24.     /  sin°  X  dx. 

sin'  a;(^  cos'  a;  —  f  cos  a;)  —  y\  sin  a;  cos  a;  +  fi^  +  ^• 
/"sin*  a;  ,  cosaj/sin^a;       sin' a;       sin  a;\    ,    a;     ,    ^ 

t/   sec  a;  2     \     3  12  8/16 

sin*  X  sec'^  x  =  sin\a;  cos^  x  =  sin*  x  —  sin°  a;. 
Hence  we  may  make  u  =  cos  x,  r  =  0,  and  v  =  sin"  a;  —  sin"  x. 

Psia'xdx        sin  a;  sin  a;         -,  ■,      ,  ,  ,        x    ,  ^ 

2b.     /  1 — -.     -, i ^ -J  log  (sec  a;  +  tan  a;)  +  (7. 

t/      cos  X         4  cos*  a;       8  cos  x      ^     ^  ^  '   ^ 

sin^  a;  _  1  —  cos^  ^'  _      ^  ^ 

cos^  a;  cos^  x  cos^  a-       cos'  x 

Hence  we  may  make  u  =  sin  x,  r  =  0,  and  v  —  cos"®  x  —  cos"^  a;. 

27.     /  cos*  a;  cosec'  x  dx.     —  ^   .   „ cos  a;  —  f  losr  tan  tt  -{-  C. 

*J  2  sm^  a;  '^     ^  2     ' 


2  sin' a-  a  -s    -    g 

X       5  sin'  a; 
3  cos'  X       3  cos  a; 


/   •   6          17  sm^T        5  sm  a;    ,  .^        •               n  ,   ^ 

28.  /  sm  a;  sec  a;  aa;.    ;r = — • — k|  a;— sma;cosa;  4-f'. 

t/  3  cos  a;       3  cos  a;       '^'-                       ^ 

Aec'a;  1/15           5     .       \ 

29.  /  ^ dx. — :^-. V-  - — r —  sm  a; 

^  sm*  X  cos^  aA3  sm  a;       3  sm  a;      2           / 

+  f  log  (sec  X  +  tan  a;)  +  G. 


208        DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

215.  Reduction  Formulas  for  Binomial  Differentials. 

These  may  be  easily  obtained  by  the  method  of  indeterminate 
coefficients. 

I.  Kequired   /  (a  -f-  hx^yx'^dx  in  terms  of 

/  (a  +  Ix'^yx^-^'dx. 

Make  w  =  a  +  bx^ ,  r  =  p,  -j-  =  nbx^~^,    v  =  x^,    s  =  r, 
and  Vj  =  :?™~^ 

Substituting  in  (F),  we  have 

x^  =  {p-{-  l)nhx''-^f{x)  +  (a  +  hx'')f'{x)  +  ^a;'"-^      (1) 

where    f{x)  =  ^a;™-"+S     and    f'{x)  =  {m  —  n -\-  l)Ax'^-\ 

Substituting  these  values  in  (1),  arranging  with  reference  to 
X,  we  find 

1  a{m  —  71  +  1) 

b{7ip  +  w  +  1)  b{np  +  m  +  1) 

.-.    /  {a  +  hx^Yx'^dx  =  —n — \  ^    ,   A 
«/   ^     '        '  bynp  +  m  +  1) 

h{np  +  m  +  1)^  ^      '        ^  ^    '^ 

By  a  repetition  of  this  formula  7n  may  be  diminished  by  any 
integral  multiple  of  n. 

II.  Eequired    /  («  +  bx'^yx'^dx   in  terms  of 


f{a  +  5a;") 


p-'^x'^dx. 

Make  w  =  «  -f-  5a;%    r  =  j:)  —  1,    s  =  j??  —  1, 

v  =  (a  +  bx")x"',       and     v,  =  a;™. 


integration:  209 

Substituting  in  (F)  and  proceeding  as  before,  we  get 

f(a  +  bx-Yx^dx  =  -^"^'(^  +  ^-^y. 

_| ^^P  f{a  +  hx-^Y-^x'\lx,      (B) 

np  -\-m  +  1  */  ^  ^    ^ 

Each  application  of  this  formula  diminishes  the  exponent  of 
a  -j-  bx"-  by  unity. 

When  m  or  p  is  negative,  we  need  formulas  for  increasing 
instead  of  diminishing  them;  hence  the  following  : 

III.  Eequired     I  {a  -\-  hx^Yx^dx  in  terms  of 

J  {a  +  lx''Yx^+''dx. 

Solving  (A)  for  J  {a  -{-  J2;")^a;™~"(^a;,  and  substituting  m  -{-n 
for  m,  we  get 

f(a  +  Ix^Y^^dx  =  ^"^'(^  +^^y+^ 
t/   ^  '  a\rti  + 1) 

-  M"P  + »  +  '«  +  !)  /-(„  _(.  u'Yz-»'ix.    (C) 

lY.  Eequired    I  [a  -\-  bx^Y'^^^^  ^^  terms  of 

y  (« +  bx^Y^^^'^'ci^- 

Solving  (B)  for  J  {a -\-  bx'^Y^^^'^dx,  and  substituting  p  -{-1 
for  p,  we  find 

/  ia  +  &a;")^a;'"^a;  = ^-^ — -f — 

t/  '  an{p  +  1) 

^^y  +  ^^  +  m  +  l    r^^^      bx^r-^'x-dx.       (D) 


210        DIFFERENTIAL  AND  INTEGBAL  CALCULUS. 

216.  The  approximate  integral  of  many  differentials  may 
be  conveniently  obtained  by  the  method  of  indeterminate  coeffi- 
cients. The  following  important  example  will  serve  to  illustrate 
the  process. 

Integrate  the  Elliptic  Differential 

dx 

^  '   Va"  —  x^ 

Comparing  this  with  (E),  we  may  make  u  —  a^  —  x^,  whence 

du 
dx 
oped  by  the  Binomial  Theorem,  gives 

2(1  8a'  16a' 

Substituting  in  {¥),  Art.  211,  making  s  =  —  l,v^=  I,  we 
have 


r  r=  —  I,  '-^  =  —  2x,  and   v  =  {a^  —  e^x'f,  which,  when  devel- 


ex       ex 


e^x" 
16t? 


xf{x)  +  {a'-xY'{^)  +  ^, 


where /(r»)  is  evidently  of  the  form /a; 
Proceeding  as  in  Art.  212,  we  find 


Ax'  +  Bx'  +  Cx. 


A  = 


96«' 

1" 


^  =  Z^«^  +  .-^'    G=^Ba^  + 


32« 


2 


4a' 


]c=i  a—  Ca\ 


s   i   ={a'-xy 

0 


+ 


5e 


^'^^-^V(S^+i;)(i)' 


96  \a 


256 


3e^      e^\  Ix 
64  "^  4  /  la 


+  «ll-4 


3e^ 
64 


5e^ 
256  ■ 


sm" 


CHAPTEE  X. 
INTEGRATION  AS  A  SUMMATION  OF  ELEMENTS. 

ELEMENTS  OF  FUNCTIONS. 

217.  Hitherto  nothing  has  been  said  about  the  magnitude 
of  differentials.  Whether  they  are  large  or  small  does  not  affect 
the  principles  which  have  been  deduced;  hence  we  may  regard 
them  as  small  as  we  jDlease.  They  are  variables  whose  limits  are 
zero. 

218.  In  the  present  chapter  increments  are  called  and  treated 
as  Elements.*  Thus  z/y  or  ^/{a:)  {=  mji  -f  mji\  Art.  24)  is  an 
element  of  the  function  y  =  f[x).  For  convenience  the  element 
///(:<■)  will  often  be  represented  by  E^,  and  the  differential  dy 
or  f'{x)dx  by  D^,,  which  may  be  called,  respectively,  the  a;th 
element  and  the  icth  differential  of  the  function  /(«).  Since 
L^  varies  as  dx  and  approaches  E^  indefinitely  as  dx  approaches 
0,  D^  is  called  the  differential  value  of  E^^  with  respect  to  dx. 

The  expression  2   \_Ex\  represents  the  sum  of  all  the  ele- 

X, 

ments  like  Ex,  or  the  sum  of  the  successive  values  of  E^,  be- 
tween the  a;-limits  x,  and  x^.  That  is,  supposing  the  increment 
of  X  to  be  always  It, 

2^^Ex\  =  Ex,  +  Ex,+h  +  Ex,+ih-\-  '  '  ■  Ex^  _  h  (or  ar,  +  («.  -  D/i), 

219.  A  Definite  Integral  Eegarded  as  a  Sum.  The  prac- 
tical importance  of  integration  consists  chiefly  in  regarding  it 

*  Because  sum  and  element  are  correlative  terms. 

211 


212        DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 


as  the  summation  of  a  certain  series.  For  example,  in  seeking 
the  area  of  a  curve,  we  conceive  it  divided  into  an  indefinite 
number  of  suitable  elementary  areas,  of  which  we  seek  to  deter- 
mine the  sum  by  a  process  of  integration.  The  solution  of  this 
fundamental  problem  is  effected  by  the  following  formula  and 
its  corollaries. 

Suppose  that  in  any  function  of  x,  a,sf{x),  we  change  x  from 
Xj^  to  x^  by  giving  to  x  successive  increments.  The  whole  change 
in  the  value  olf{x),  viz.,/(a;J  — /(«j),  must  be  the  sum  of  the 
partial  changes  produced  by  the  increments  given  to  x.     That 


IS, 


f{^.)-f{^.)  =  2\4f{^)l 


(A) 


or  (Art.  208)  TX  =  ^''[^4  •     • 

Formula  (A)  is  not  only  an  expression  of  the  simple  fact 
that  the  whole  is  equal  to  the  sum  of  its  parts  or  elements,  but 
it  signifies  that  the  integral  of  the  differential  value  of  an 
element,  between  certain  limits,  is  the  sum  of  the  successive 
values  of  that  element,  between  the  same  limits. 


As  an  illustration  of  (A)  let  us  consider  the  area  of  EGQF^ 
where  OE  —  x^  and  OG  —  x^.  Suppose  y—f'{x)  to  be  the 
equation  of  AFQ,  where  x  =  OB  and  y  =  BP. 


INTEGRATION  A8  A  SUMMATION  OF  ELEMENTS.    213 

Let  u  =  the  area  of  OBPA,  and  let  BC  {—  h)  be  an  element 
of  X,  then  BCDF  =  du  =f'{x)dx  =  D^,  and  BCP'P  =  D:c  + 

au  =:  Ex' 

(a)  Evidently,        EGQF=    r^I),^,  Art.  20S.     .     .     .     (1) 

(b)  Divide  EG  {=  x^  —  Xj)  into  n  parts,  each  equal  to  7i,  and 
draw  the  ordinates  a^d^,  a^d^,  a^d^,  etc.,  then 

EGQF=Ed,  +  a,d,-^a,d,+,    ....     (2) 

in  which  Ed^ ,  a^d^ ,  a^d^,  etc.,  are  the  successive  values  of  Ex,  as 
X  increases  by  li  from  x^  to  x^.  That  is,  Ed^=Ex^ ,  a^d^=Exi-\-h, 
a^d^  =  Exy  -\-2h,  etc.     Hence  (2)  may  be  written 

EGQF=2''\E:,:\ (3) 

Now,  equating  (1)  and  (3),  and  we  obtain  formula  (A). 

In  further  illustration  of  formula  (A),  let  us  show  that  the 

signification  which  it  expresses  is  true  of   /     ^x^dx. 

ia)  r^Zx'dx  =  \x'  +  of'  =  x;  -  x^\      ...     (4) 

(b)     Since  f{x)  =  3x',  E^  =  2,x%  +  ^xh'  +  h\  Art.  207, 
Ex,       =2,x^li-^%xji'  +  ¥; 
Ex,  +  h  =3x^Vi  +  9xJi'-h7h'', 
Ex,  +  2h  =  3a:,  Vi  +  15xji"-  +  19A'; 


Ex,+in-m=  3a:,7i  +  dx^{27i  -  l)h'  +  (3w^--  3n  +  l)h\ 

Taking  the  sum,  remembering  that  a;,  +  nh  =  x^ ,  and,  by 
Algebra,  3  +  9  +  15  +  .  .  .  d{2n  -1)  =  3w^  and  1  +  7  +  19  + 
.  .  .  (3;i*  —  dti  +  1)  =  n%  we  have 


^""'[^J  =  Snhx^'  +  Sinh)%  +  {nhy 


0 

Xl 

=  {x^  +  nhy  -  X,'  =  x;  -  x^\     ...     (5) 


214        DIFFERENTIAL  AND  INTEGRAL  CALCULUS 

Comparing  (4)  and  (5),  we  see  that  the  results  of  the  opera- 
tions  indicated  in  (A),  when  applied  to    /     'ix^dx,  are  the  same. 

220.  Formula  (A)  is  also  true  when  x^  —  x^  {=  EG)  is  not 
divided  into  equal  parts. 

Let  us  suppose  x„  —  x^  to  be  divided  into  the  following  equal 
or  unequal  positive  parts : 

a  —  x^,     h  —  a,     c  —  h,  .  .  .  I  —  h,     x^  —  l, 

the  sum  of  which  is  evidently  x^  —  x/,  then  we  have  identically 

/    ydx—    I   y  dx -\-    I    y  dx -\-   /  y  dx -\- .  .,  .    I   "^ydx,      (1) 

i/x\  f-'xi  *J a    '  fJb  t/k 

in  which  a  —  x^,  b  —  a,  c  —  h,  etc.,  may  be  considered  the  suc- 
cessive values  of  z/.r  and    /    y  dx,   /   ydx,  etc.,   the  correspond- 

ing  successive  values  of  E^.  Hence  (1)  is  the  general  significa- 
tion of  (A),  which  the  student  may  easily  illustrate  with  a 
figure. 

221.  A  Defi.nite  Integral  Regarded  as  the  Limit  of  a  Sum. 

In  Fig.  43  let  us  suppose  n  to  increase  and  h  to  decrease,  nh 
being  always  equal  to  UG.  Since  the  limit  of  E^  -^  D^,  o.^  h 
approaches  0,  is  unity,  the  sum  of  all  the  rectangles  like  D^ 
approaches  indefinitely  the  constant  sum  of  all  the  elements 
like  E^.     Therefore 


limit  y?'^ 


A 


=  ^^" 


E, 


Substituting  in  (A),  Art.  219,  we  have 

limit  ^^' 


t/Xi 


i).  =  l™M'U. ,(B) 


That  is,  the  definite  integral   /     Dx  is  equal  to  the  limit  of 

the  sum  of  all  the  successive  values  of  D^,  as  x  increases  by  h 
from  X,  to  x„. 


INTEQBATION  AS  A  SUMMATION  OF  ELEMENTS.     215 

For  example,  let  lis  find  the  valiie  of  /     ^x^dx. 
Since         Dx  =  "^xVi,    we  have 
Dx,  =  3x;h, 
Dx,  +  h  =  ^x;h  +  6xji'  +  Zh% 


Dx,  +  (n-\)h  =  ^x^%  +  6(w  -  Vixji'  +  ^in  -  \yii\ 
Taking  the  sum,  remembering  that 

0  +  6  +  12  +  .  .  .  6{w  -  1)  =  3(w'  -  n), 

and  0  +  3  +  12  +  27  +  .. .3(.^-lr=(^-^^K^!^^:ll), 
we  have 

^"'[Z),]  =  Znhx;-  +  3(7^^  -  n)xji'  +  0^  -  ^){n){2n  -  1)^^, 

^  2 

=  ^{nh)x^'  +  3(wA)=a;,  -  d{nJi)xJi  +  (m/^^'  -  |(«/0'A  4-  &)7i^ 

Now,  making  w^  =  x^  —  .«, ,  and  then  passing  to  the  limit 
by  making  h  =  0,  we  have 

^'^'^Q^ZlI).]  =  3(.^,  -  ^.W  +  3(0;,  -  x,)\  +  {X,  -  a;.)' 


which  is  evidently  equal  to     /     'Hx'dx. 


222.  It  is  important  to  observe  that,  whether  an  integral  be 
regarded  as  a  sum,  or  the  Ihnit  of  a  sum,  integrating  is  equiv- 
alent to  two  distinct  operations : 

(a)  If  a  sum,  as  in  Art.  219,  integrating /'(a;)fZa;  is  equivalent 
to  (1)  increasing  the  differential  f\x)dx  by  the  acceleration  au 
to  obtain  the  element  E^,  and  (2)  finding  the  sum  of  the  succes- 
sive values  of  E^. 


216         DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

(b)  If  the  limit  of  a  sum,  as  in  Art.  221,  integrating/' (a;) ^a; 
is  equivalent  to  (1)  finding  the  sum  of  the  successive  values  of 
f'{x)h,  and  (2)  taking  the  limit  of  the  sum,  as  li  approaches  0. 

In  case  {a)  all  the  quantities  involved  are  finite;  but  in  case 
{h)  the  limit  of  each  part  is  0,  and  the  limit  of  the  number  of 
parts  is  co  .  Both  methods  have  their  advantages,  and  hence  both 
will  be  employed,  more  or  less,  in  the  applications  which  follow. 

Just  here  the  student  may  profitably  read  Art.  238,  which 
offers  a  simple  illustration  of  the  significations  and  practical  im- 
portance of  formulas  (A)  and  (B). 

APPTJCATIONS   TO   GEOMETRY.* 

223.  Lengths  of  Curves.— I.  Rectangular  Co-ordinates. 

To  find  the  length  (s)  of  the  arc  APQ  between  the  limits 
OB  =  x^  and  OG  =  x^.     (Fig.  44.) 

a. 


Fig.  44. 


Here  E^  =  Pd,  and  D^c  =  Pt  =■  y\-\-  (^)'/i.  Art.  33,  and 


*  The  previous  applications  of  Calculus  to  Geometry,  Arts.  63,  64,  65, 
66,  were  limited  to  the  most  elementary  rules  for  integration  ;  in  this 
chapter  it  is  our  purpose  to  extend  these  applications  by  the  more  advanced 
methods  of  integration  with  which  the  student  is  now  familiar,  and  in  doing 
so  to  impress  upon  him  the  important  principle  of  integration  as  a  summa- 
tion. 


INTEGRATION  AS  A  SUMMATION  OF  ELEMENTS.    217 

x^  limif      •'''2  P^i 

Pd,  +  dA  +  dA  +  etc.  =  2^[E.-\  or  j^^^2^[D,-\=J^^  D,. 


EXAMPLES. 

X^  1 

1.  Find  the  length  of  the  arc  of  the  curve  y  =  —  +  -—  be- 

tween  the  limits  ic^  =  1  and  x^  =  2. 

(Z.V  _  g-^  -  1       (         dir\i_\+x'  _  ds 
^^^^       dx  ~     2X-'     '     \    "^  dx'l  ~~     2x'  dx 

'I  +  x'\  -.  r  dx  , 

^'  1 

r       1         a;'~l^ 
— —  _j__     —  j_i 

~  L       '^^         6Ji~  ^^' 

2.  Eectify  the  parabola  y^  =  4aa-,  using  the  formula 

dy       ^  \dyl^ 


x^dx 

1  t^'  1  •-/ 1 


rfa;         ?/  ds       a/  y'^    ,   -,        1    . /-5— ; — T~t 

d^  =  ^'     •••^-^4^  +  ^  =  2^^'^  +^"- 

= -^ — h  « log    •^— ^ z: —^.  Art.  214,  Ex.  10. 

4a  °  \  2a  / 

3.  Eectify  the  curve  y  =  log  (x  +  4/a;2  —  1). 

4.  Rectify  the  ellipse  y^  =  {1  —  e^){a'  —  x^). 


dii  ,,        ^.x  X  VI  —  e' 


^^  1/  Va'  -  x"" 


218         DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 


To  find  the  length  of  a  quadrant,  we  must  integrate  between 
the  limits  0  and  a;  hence 


=  £  /l  +  '!!_  5^  dx  =  £  ^a^  -  e^x^ 


dx 


a  —  X' 


Va'-af 


^(.       1   2         1.3     ,        1.3\5     e        ^     ,       A   ^    o-.^ 


5.  Rectify  the  cycloid  a;  =  r  vers~^  -_  —  V"2ry  —  ^\ 
Here  dx  =  —-^         _ ; 

^        2r_?/  —  2/  2r  —  y 

s  =  J'{2?f{2r  -  yy'dy  =  -  3  |/3r(2r  -  ?/)  +  C'- 

If  we  estimate  the  arc  from  the 
point  B  where  y  =  2r,  we  shall 
have,  when  s  ^  0,  y  =  2r;  hence, 

0  =  0  +  C,     .:  C=0,  and 


s  —  —  2  V2r{2r  —  y). 


Since  BC  =  2r  and  BE  =  2r  -y,BG=  V2r{2r  -  y). 

BD=s  =  2B0; 

or  the  arc  of  a  cycloid,  estimated  from  the  vertex  of  the  axis,  is 
equal  to  twice  the  corresponding  chord  of  the  generating  circle; 
hence  tJie  entire  arc  BDA  is  equal  to  tivice  the  diameter  BC,  and 
the  entire  curve  ADBH  is  equal  to  four  times  the  diameter  of 
the  generating  circle. 

6.  Kectify2/  =  ^  +  2^.  ^  =' ^  "  "^  +  ^- 


INTEGRATION  AS  A  SUMMATION  OF  ELEMENTS.    219 

Note. — The  value  of  C  depends  on  the  point  from  which  s  is 
measured.  ThuS;,  if  s  is  estimated  from  x  =  1,  then  s  =  0  when 
X  =  1,  and  we  have  0  =  ^-^  —  -^  +  C;  that  is,  C  =  ^^. 


^n+l 


7.  Eectify  y  =  + 


1 


8.  Rectify  ij  =  -—■—  — -^ -.  s  =  w^  +  ~ \-  C. 

9.  Eectify  y  =  ^  log  {x'  +  3a;  +  2). 

10.  Eectify  y  =  ^x^  —  log  x.  s  =  |a;^  +  log  x-\-  C. 

A  curve  is  said  to  be  rectifiaMe  when  its  length  can  be  ex- 
pressed in  finite  terms  by  aid  of  the  algebraic  and  elementary 
transcendental  functions. 

224.  II.  Polar  Co-ordinates.  To  find  the  lengtli  {s)  of 
the  arc  APQ  between  the  limits  6^  =  AOP  or  r^  =  OP,  and 
e^  =  AOQov  r^=OQ.     (Fig.  46.) 

Here  Ee  =  Pd^  and  De  =  Pt  =  Vr'^i^^dd,  Art.  97; 
hence 

Pd^  +  d^d,  +  d/l,  +  etc.  =  :S^[Eb] 

limit   ^^Vn  1  /'^Vn  n 

11.  Eectify  the  spiral  of  Archimedes,  r  =  aO. 

Here  ^=-',     .:  s  =   P (l  +  ~]'dr  =  -  /" («'  +  r'Ydr 
dr      a'  t/o    \     '  ay  a ^o   ^      '      ^ 

=     ^    J +  -  log  — ^— ^^-.  Art.  214,  Ex.  10. 

9,/7  '       9,  *=  /7  ^ 


220        DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 


Let  this  result  be  compared  with  the  length  of  the  parabola, 
Ex.  2. 

12.  Find  the  length  of  the  logarithmic  spiral  log^  r  =  6. 

r    '      '  '   dr~  r' 
and  s  =  J^   {1  +  m')hlr  =  r{l  -\-  vi^)\ 


Here 


13.  Find  the  length,  measured  from  the  origin,  of  the  curve 

a"  —  x^ 


3/  =  « log 


,      a  4-  X 
a  log X. 


a'  ^  a  —  X 

14.  Find  the  entire  length  of  the  arc  of  the  hypocycloid 

^^  ~\-  y^  =  a%  6«. 

15.  Find  the  entiz'e  length  of  the  cardioid  ?•  =  a{l  —  cos  6). 

Sa. 


16.  Find  the  entire  length  of  the  curve  7^  =  a  sin' 


6       Syr  a 


INTEQBATION  AS  A  SUMMATION  OF  ELEMENTS.    221 

17.  Find  the  length,  between  x  =■  a  and  x  =  b,  oi  the  cnrve 

18.  Find  the  length  of  the  tractrix,  measured  from  (0,  a),  its 
differential  equation  being 

ds  a  ,      a 

-J—  = .  o  log  — . 

dy  y  ^  y 

19.  Find  the  length  of  the  arc,  measured  from  the  vertex,  of 
the  catenary 

20.  Find  the  length  of  a  quadrant  of  the  curve 

a)   '^\b  I        '  a  +  b      ' 

225.  To  find  tlie  equation  of  a  curve  when  its  length  is 
given. 

1.  Find  the  equation  of  a  curve  whose  length  is 

x^ 
s  =  ilogx-\-j. 


ds 
dx 


_  1 4-  ^\    .  i// ^■^  y  _ -,  _  1  —  ^"^  _  <^^ 

~      2x     '     ' '        \dx )  ~     2x     ~  dx 


I  —  x^ 
Hence,    dy  =  — ^ — dx,    and    y  =  i  log  x  —  ^x"^  -\-  G. 


EXAMPLES. 
Find  the  equations  of  the  curves  whose  lengths  are : 

2.  .  =  -  ^,  +  |\  y  =  s-'^  +  C.      See  Note,  p.  219. 


222        DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 
-1> 


4.  s  =  i  log  (tan  x) 
1 


2/=log(a.--l)+C'. 
y  =  ^ log  (sin  2x)  -\-  C. 


5.  s  =  - 


^1  + 


2{n-  IJx"-^  ^  3(w  +  l) 


y  =  s 


w+1 


+  C'. 


226.  Areas  of  Curves.— I.  Rectangular  Co-ordinates.     To 

find  the  area  of  the  surface  between  a  given  curve,  the  axis  of  x 
and  two  ordinates  whose  abscissas  are  x^  and  x^ ,  we  have.  Art. 
219, 

u=    pydx  =  2l\E^] (E) 

For  a  definite  area  between  the  curve  and  axis  of  y,  we  have 
u  =    r\cdy  =  ^JEyl (F) 

EXAMPLES. 

X^  1 

Find  the  area  of  the  curve  w  =  ——  -4-  -—  between  the  limits 


16 


=  1  and  x^  =  2. 


2.  Find  the  area  of  the  circle  y""  =  a"  —  x''. 

nx 

Area  of  OB  PA  =    I   ydx 

=    /    (a'  —  a;')'^a;  =  -^ — ^  +  —  sm  ^  - 

t/o  2  2  a 

(Ex.  17,  Art.  96) 


A'  B      C 

Fig.  47. 


(O^K^  +  (0P)(^;^  ^P)  =  „ea.OBP  +  area  OP^. 


or 


mTEGBATION  AS  A  SUMMATION  OF  ELEMENTS.    223 
To  find  the  area  of  the  quadrant  OCA  we  have 

/    («'  —  x'fdx  =  ^TTa". 

The  value  of  tt  is  given  in  Art.  129. 

3.  Find  the  area  of  an  ellipse,  a't/"  =  o'V  —  Ifx^. 

11=-  Va"  —  a;" ;     .\  u=  ~      Va"  —  x" dx. 
a  a^ 

;j  (the  area)  =  -  I    Va"  —  x^  dx  =  ^Ttab; 

entire  area  =  Trab. 

4.  Find  the  area  of  the  hyperbola,  a'/y^  =  b'^x'  —  a^b\ 

y  z=  -  Vx"  —  a";     .'.  u  =  —  I  \/af'  —  a''  dx^ 

bx(x''  —  a^)'      ah  ^      /     ,    .,— ^x    ,    ^ 

u  =  -^-^ Y  ^^^  ^^  +  ^^  "  ""-^  +  ^• 

To  find  C,  we  know  that  when  x  =  a,  u  =  0;  hence 

0  =  -  ^   log  rt  +  C;     .:  C  =  Y  log  «• 

Substituting  this  value  of  C,  and  making  l/a;^  —  a^  —  ^, 


we  have 


Area  of  hyperbola  =  "y  -  y  log  ^|  +  |-j. 

5.  Find  the  area  of  the  surface  between  the  arc  of  the  pa- 
rabola y'^  =  4:ax  and  the  axis  of  y.  ^xy. 


224        DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 


2tZl.  It  is  often  convenient  and  suggestive  to  regard  a  defi- 
nite integral  like 

r*y{dx)  =  EGQF       (Fig.  43) 

as  signifying  that  "the  ordinate  y 
(or  generatrix  PB),  moving  perpen- 
dicularly to  the  axis  of  x  from  a;=a;, 
to  ic  =  x^,  generates  EGQF. 

Thus,  let  it  be  required  to  find 
the  area  of  the  surface  between  the 
parabola  if  =  4ft«  and  the  straight 


Fig.  48. 
line  y  =  X.     We  at  once  have 


OPDG  = 


t^  =  OE 

{PC)dx 


{Viax  —  x)dx  =  fa* 


This  method  can  be  employed,  with  equal  facility,  in  finding 
the  volumes  of  many  solids,  in  which  case  the  generatrix  is  a 
surface. 

6.  Find  the  area  of  one  branch  of  the  cycloid. 

V2ru  —  y'  ^    V2ry  —  y' 


'ry  -  y 
For  the  area  of  one  branch  we  have 


u  =  % 


^^^ —  =  ^nr\        (Ex.  13,  Art.  214) 


0     ^"^ry  -  y 
which  is  three  times  that  of  the  generating  circle. 

-.6  & 

7.  Find  the  area  of  the  curve  xy  =  a.  [«]    =  a  log  -. 

8.  Find  the  area  of  the  curve  x^y  —  a-  +  1  =  0  between  the 
a;-limits  1  and  2.  i- 

9.  Find  the  area  of  both  loops  of  a*y''  —  a'^x"  —  b'^x*,       %ab. 


INTEGRATION  AS  A  SUMMATION  OF  ELEMENTS.    225 

10.  Find  the  area  of  both  loops  of  the  curve  a^y'^  =  a^x-  —  x*. 

4 

"J- 

11.  Prove  that  the  area  of  the  curve  a^^y^  =  {a^  —  x^)x^,  be- 
tween the  rc-limits  0  and  a,  is  the  same  for  all  values  of  a. 

228.  II.  Polar  Co-ordinates.  In  Fig.  46  the  area  of  POD 
[z=  h-"dd,  Art.  35)  is  the  differential  value  of  the  element  POc?,; 
therefore 

fj'hr'dd  =  POd,  +  d,Od,  +  d,Od,  +  .  .  .  d„_^OQ, 

where  d  ^  AOP  and  0^  =  AOQ. 


=  if 


r\W  =  area  POQ. (G) 


12.  Find  the  area  of  the  spiral  of  Archimedes,  r  =  ad. 

a  ^  2a  t/o  Qa 

CoK.  I.  If  fl  =  -r— ,  as  is  usual,  u  =  irrr^. 
Ztv 

If  r  =  1,  or  ^  —  2;r,  u  =  ^n,  which  is  the  area  described  by 
one  revolution  of  the  radius  vector. 

If  r  =  2,  or  6^  ==  4;r,  u  =  ^tt,  which  is  the  area  described  by- 
two  revolutions  of  the  radius  vector,  which  includes  the  first 
spire  twice;  hence  the  area  of  the  entire  spire  is  |;r  —  ■J;r  =  ^tt. 

13.  Find  the  area  of  the  hyperbolic  spiral,  r  =  —. 


a'dd 


26''      •■•    W.  ~     ~2F+^ 


a' 


20      28' 


14.  Find  the  area  of  the  logarithmic  spiral  0  =  log„  r. 

,^       mdr  , 

au  =  ;     .'.  when  m  =  1, 

du  =  — ^    and    2i  =  ^r*; 


226         DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

that  is,  the  area  of  the  natural  logarithmic  spiral  is  equal  to  one 
fourth  the  square  described  on  the  radius  vector. 

15.  Find  the  entire  area  within  the  hypocycloid 

X^  +  ^*  —   <^^^-  %7tcC' 

16.  Find   the  area  of  the   surface   between    the    parabola 
x^  =  4fl!?/  and  the  witch 

17.  Find  the  entire  area  of  the  cardioid 

r  =  a\\  —  cos  B).  \7ta^. 

18.  Find  the  area  of  a  loop  of  the  curve 

a;^  +  _?/*  =  a^xy.  \7ia^. 

19.  Find  the  area  of  the  loop  of  the  curve 

a^y-  =  x\l)  +  x). 


32*' 


105a' 

20.  Find  the  area  included  between  the  axes  and  the  curve 

21.  Find  the  area  between  the  curve  x'^y^  +  ^^  =  (^^^^  aiid 
one  of  its  asymptotes.  2a\ 

22.  Find  the  area  of  the  loop  of  the  curve 

y^  ■{-  ax^  —  axy  =  0.  gJ„-rt\ 

23.  Find  the  area  of  the  three  loops  of  the  curve 

r  =  asindd.     (See  Fig.  38.)  i7ra\ 

24.  Show  that  the  whole  area  of  r  =  a  (sin  20  -{-  cos  26)  is 
equal  to  that  of  a  circle  whose  radius  is  a. 

229.  Areas  of  Surfaces  of  Revolution.     By  Art.  34  the 
differential  value  of  the  a;th  element  of  surface  is 


^"■'{^  +  <¥)*'^^'- 


INTEGRATION  AS  A  SUMMATION  OF  ELEMENTS.     227 
therefore 


' = ^  V.  ^(^ + '^J'^'-  =  ^:^:t^^]- 


(H) 


EXAMPLES. 


1.  Find  the  area  of  the  surface  generated  by  revolving  the 
;   of   the  curve   ; 
x^  =  2  and  x^  =  4. 


a;'         1 
arc   of  the  curve  y  =  t^  +  h^  about  the  axis  of  x,  between 

lb  /iX 


'      ^  \dxl      dx       4  ^  x' 


«=2-/  (^+oM?+!>- 


16   '    2a;VV4    '   x 


=H 


256  "^  Te  ~  4^^ 


+  (7    =257^^4-^. 


2.  Find  the  area  of  the  surface  of  a  prolate  spheroid,  the 
generating  curve  being  the  ellipse  a^y'^  =  ¥{a^  ~  y^)- 

b 


V  =  —  Va"  —  x'',    and     ds  =  y 


a  —  ex 
a"  —  x^ 


■dx. 


.:  Area  =  2   /     27tyds  =  An-   I     (a'  —  e'x'^fdx 
Jo  ""Jo 


=    4:71- 


xia'  -  e'x'f  ,    a'     .     ,  ex-]'^ 

— ~  -\ sm"  — 

2  ^  2e  a  J  „ 


27rab   .     . 

=  27tb  -\ sm"^  e. 

e 

3.  Find  the  area  of  the  surface  generated  by  the  revolution 
of  the  cycloid  about  its  base. 


/•2r  /»2 

Area  =  2    /     27tyds  =  4t7i  \/2i'    I 

i^'^O  t/-  0 


_ydy 
0      V2r-y 


228         DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

,3r 


=  4.7r  ^2r\-  |(4r  +  .v)(2r  -  yy~] 
L  J, 


64 

4.  Find  the  surface  of  the  paraboloid  between  the  limits 
X  =  0  and  x  =  a,  the  generating  curve  being  y'  =  4fla;. 

i(|/8  -  l)d,7ra\ 

5.  Find  the  surface  generated  when  the  cycloid  revolves 
about  the  tangent  at  its  vertex.  ^-tti-"^. 

6.  Find  the  surface  generated  by  the  revolution  about  the 
axis  of  X  of  the  portion  of  the  curve  y  =  &^,  which  is  on  the  left 
of  the  axis  of  y.  n^V^  +  log  (1  +  4/2)). 

7.  Find  the  entire  surface  of  the  oblate  spheroid  produced 

by  the  revolution  of  the  ellipse  a^y^  +  Ifx^  =  a'b"  about  its  minor 

axis.  „     ,    ,      &%      1  + « 

27ra  +  7t—  log-—! — . 

8.  Find  the  surface  generated  when  the  cycloid  revolves  about 
its  axis.  STtr^TT  —  A). 

9.  Find  the  surface  generated  by  revolving  the  arc  of  the 
cardioid  r  =  «(1  —  cos  0)  about  the  initial  axis. 

S  =  27tJ*yds  =  27tfr  sin  d  \/dr'  +  r^dd\  ^-na\ 

10.  Find  the  surface  generated  by  the  revolution  of  a  loop  of 
the  lemniscate  ?•'  =  a"  sin  20  about  the  polar  axis.  27ia^. 

230.  Volumes  of  Solids  of  Revolution.  In  Art.  32  the 
volume  of  the  cylinder  {ny'^dx)  generated  by  the  revolution  of 
the  rectangle  BCDP  is  the  differential  value  of  the  a;th  element 
of  volume;  therefore 

.  =  ^;[^J    or     f2l^^\nfk-]  =  ^£ydx.        (J) 


INTEGBATION  A8  A  SUMMATION  OF  ELEMENTS,    229 


EXAMPLES. 

1.  Find  the  volume  of  a  paraboloid,  the  generating  curve  be- 
ing the  parabola  y'^  =  4iCix. 

V  ^=  7t  I    y'^dx  =  47ra  /    xdx  ■=.  'inax^. 

When  the  curve  is  revolved  about  the  axis  of  y,  we  evidently 
have 

V  =  7T  I  x^dy. 

2.  Find  the  volume  generated  by  revolving  the  surface  be- 
tween the  parabola  y  =  -\-V4iax  and  the  axis  of  y  about  that  axis. 

[v-\\  =  nfxhly  =  nf-^^dy  =  ^  =  \nx^y. 

That  is,  the  entire  volume  is  one  fifth  of  the  volume  of  the 
circumscribing  cylinder;  therefore  the  volume  generated  by  the 
surface  of  the  parabola  in  revolving  it  about  the  axis  of  y  is  four 
fifths  of  that  of  the  circumscribing  cylinder. 

3.  Find  the  volume  of  the  solid  generated  by  the  revolution 
of  the  cycloid  about  its  base. 

^^=        y'^^'       ;    :.dv  =  nfdx=      ^-^'^^^ 


^^ry  —  y"^  V2ry  —  y^ 

To  obtain  half  of  the  volume,  we  must  integrate  between  the 
limits  y  =  0  and  y  —  2r. 

J^     V2ry  -  y' 

That  is,  V  =  f  of  the  circumscribed  cylinder. 
4.  Find  the  volume  generated  by  revolving  the  ellipse  AA' 
about  the  tangent  X'Xas  an  axis.     (Fig.  49.) 

Let    OA  =  a,    OY=b,    O'B  =  x,  BP  =  y,  and  BP' =  if; 


tlisn        y  =  -{a  +  Vci''  —  x"),    and    y'  =  -{a  —  Va"  —  x"). 

0»  CI 


230        DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 
Supposing  P'n  =  dx,  the  volume  generated  by  P'm,  viz.. 


n{y^  —  y''')dx    or     4:7--  \/a'' 


JU     Cif^y 


-X- 


/ 

3 

^^ 

~~~^^-\ 

m 

A'i 

^ 

0 

p^ 

yi'          .     - 

O' 

Fig.  49. 


is  the  differential  value  of  the  volume  of  the  element  generated 
by  P'cdP,  with  respect  to  dx. 


f 


4:7C-  Va'  -  x\lx  =  27r'aF,      (Ex.    17,  p.    56) 


which  is  the  entire  volume,  being  the  sum  of  the  volumes  gen- 
erated by  revolving  all  the  elements  like  P'cdP  between  x  =  —a 
and  a;  =  a,  or  the  limit  of  the  sum  of  the  volumes  generated  by 
all  the  rectangles  like  P'nmP, 

5.  Find  the  volume  of  the  closed  portion  of  the  solid  gener- 
ated by  the  revolution  of  the  curve  {y"^  —  b'^Y  =■  a^x  around  the 
axis  of  y.  256  nh^ 

6.  Find    the    volume    generated    by   revolving    the    curve 

{x  —  4ia)y^  =  ax{x  —  3a)   about  the    axis    of  x,   between    the 

a;-limits  0  and  3a.  ^«%,k       -.^.i      r,\ 

—  (15  -  16  log  2). 

7.  Find  the  volume  generated  by  revolving  the  cycloid  round 
the  tangent  at  the  vertex.  n'^r^. 

8.  Find  the  volume  and  surface  of  the  torus  generated  by 
revolving  the  circle  x"^  '\-  {y  —  by  =  a^  ^bout  the  axis  of  x. 

^Trcvb  and  An^ab. 


INTEGRATION  A8  A  SUMMATION  OF  ELEMENTS.     231 

9.  Find  the  entire  volume  and  surface  generated  by  revolv- 
ing the  hypocycloid  x^  -{-  y^  =  (f  about  the  axis  of  X. 

and 


105  5 

10.  Find  the  volume  generated  by  the  curve  xy'^  =  4:a^{2a  —  x) 
revolving  about  its  asymptote.  47r^a\ 

11.  One  branch  of  the  sinusoid  //  =  i  sin  —  is  revolved  about 
the  axis  of  x;  find  the  volume  generated.  iTr'^ab^. 


SUCCESSIVE   INTEGRATION. 

231.  A  Double  Integral  is  the  indicated  result  of  reversing 
the  operations  represented  by  . 

Thus,  if  -^ — Y~  —  ^y^^     then    u  =    I    I  xy"dy  clx, 

which  indicates  two  successive  integrations,  the  first  with  refer- 
ence to  x,  regarding  y  as  constant,  and  the  second  with  reference 
to  y,  regarding  x  as  constant. 

Thus,  J  J  xy"-dtj  clx  =J  [^  +  C^y'^dy 

where  C  and  C^  are  the  constants  of  integration. 

232.  Definite  Double  Integrals.     Here  both  the  integra- 
tions are  between  given  limits. 


pc     Pa 

For  example,  /     /    x^y'^dx  dy. 


This  notation  indicates  that  the  integrations  are  to  be  taken 
in  the  following  order : 


232        DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

pc    pa  pel    pa  \ 


«|L),,=  g  (,._,.). 


That  is,  as  dy  is  written  last,  the  y-integration  is  taken  first. 
The  limits  of  the  first  integration  are  often  functions  of  the 
second  variable. 
For  example, 

dx dy  =    j     (  Vx)dx  =  |a=. 


ff 

,7  0       ty'  0 


As  another  example, 


rdr  dO  ■=. 


h   \b 


dr  =  A^>^ 


233.  A  Triple  Integral  is  the  indicated  operations  of  three 
successive  integrations,  for  which  the  notation  is  similar  to  that 
of  double  integrals.     Thus, 


dx. 


EXAMPLES. 


Eind  the  following : 

{x"^  -\-y'')dxdy. 


l,ah{ce  +  b% 


l7Ta\ 


INTEGRATIOW  AS  A  SUMMATION  OF  ELEMENTS.    233 


•a 


3.  3a 


♦^a^  -  ax 

clx  dz 


^    ^^       Vax  —  x" 

X2  +  ?/2 


-'a      /'V* 


ax  —  x^ 


a 


4. 


dx  dy  dz. 


4a'. 


f  ;ra  . 


0     ty  0 


AREAS  OF  SURFACES  DETERMINED  BY  DOUBLE  INTEGRATION. 
234.  Plane   Stirfaees — (a)  Rectangular  Co-ordinates.     In 
the  formula   u  =   j  y  dx,  Art.  226,  we  may  make   y  =  J  dy, 
which  gives 

u  =        I  dxdy (J) 


p^ ---^^ 

py^ 

D                              ^N. 

/  q. 

n                                       \ 

B         C 


Fig.  50. 


Art.  228,  we  may  make  -  =   I  rdr,  and  write 
u  —   j    j  rdddr.    .    . 


(K) 


234        DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

As  illustrative  examples  let  us  employ  (J)  and  (K)  in  finding 
the  area  of  a  circle  whose  radius  is  a. 

Let  {x,  y)  be  the  co-ordinates  of  the  point  m,  and  {x  -\-  dx, 
y  -\-  dy)  of  the  point  p,  then  mn2)q  =  dxdy. 

Kegarding  x  as  constant,  we  have 

y=BP 

BCDP  =  2  [mnpq} 

j/=0 


=  c?a;  /      dy  =  V2ax  —  x^dx. 


Again,  since  V'2ax  —  x^dx  is  the  differential  value  of  BP' 
with  respect  to  dx,  we  have 

2a  /»'-''*     

2    [BF']  =   /     V2ax  -  x'dx  =  ^Tia'  =  area  of  OKP. 

0  ^0 

{b)  Let  (r,  d)  be  the  co-ordinates  of  m,  and  (r  +  dr,  d  -\-  dd) 
of  p',  then  mn  =  dr,  mq  —  rdd,  and  rdddr  (  =  area  of  mnpq)  is 


Fig.  51. 


the  differential  value  of  the  element  mnp'q'  with  respect  to  dr. 

Therefore,  regarding  B  as  constant,  we  have 

OP  /.'^a  cos  ^ 

2     Imnp'q'^  =    /      rt^^^/r  =  2a'  cos^  6^^^  =:  OPP/. 
0  '^ 


INTEGRATION  AS  A  SUMMATION  OF  ELEMENTS.     235 

Again,  since  the  area  of  OPP^'  is  the  differential  value  of 
the  element  OPP',  with  respect  to  dO,  we  have 


2\0PP']  =   f  ici-  cos'^  BdB  =  iTta' 
which  is  one  half  the  area  of  the  circle. 


EXAMPLES. 

1.  Find  the  area  (1)  of  a  rectangle  by  double  integration; 
(2)  of  a  triangle. 

2.  Find  the  area  between  the  parabola  _z/^  =  ax  and  the  circle 

3.  Find  by  double  integration  the  entire  area  of  the  cardioid 
r  =  a{l  —  cos  8).  ^Ttn'^ 

4.  In  a  similar  manner  find  the  entire  area  of  the  Lemniscate 
?•«=«"  cos  2^.  n\ 

5.  Find  the  whole  area  between  the  curve  xy'^  =  Aa"^  {2a  —  x) 
and  its  asymptote.  4;ra'. 

236.  Surfaces  in  General.— To  find  the  area  (=  S)  of  a 
surface  whose  equation  is  f{x,  y,  z)  =  0. 

Let  {x,  y,  z)  be  the  co-ordinates  of  any  point  P  of  the  surface, 
and  [x  +  dx,  y  +  dy,  z  +  dz)  the  co-ordinates  of  a  second  point 
Q  very  near  the  first  (Fig.  52).  Draw  planes  through  P  and  Q 
parallel  to  the  planes  XZ  and  YZ.  These  planes  will  intercept 
a  curved  quadrilateral  P^  on  the  surface;  its  projection  jt?^',  a 
rectangle,  on  the  plane  of  XY;  and  a  parallelogram  p'q',  not 
shown  in  the  figure,  on  the  tangent  plane  at  P,  of  which  pq 
is  the  projection.  The  area  of  2^'q'  =  dS,  since  it  is  the  dif- 
ferential value  of  PQ  (=  AS)  with  respect  to  dx  and  dy. 

The  projection  of  j/q'  on  XY  is  dxdy\  similarly  the  projec- 
tions of  p'q'  on  XZ  and  YZ  are  dxdz  and  dydz;  hence,  denoting 


236         DIFFERENTIAL  AND  INTEQBAL  CALCULUS. 

the  angles  between  the  plane  of  p'q'  and  XT,  XZ  and  YZ  by 
a,  §  and  /,  respectively,  we  have 

cos  adB  =  dxdy, (1) 

cos  ftdS  =  dxdz, (2) 

cos  yds  =  dydz (3) 


Fig.  53. 


Squaring  (1),  (2),  (3),  and  adding,  remembering  that 
cos'  a  4-  cos*  /?  +  cos'  y  =  1, 


we  have 


hence, 


{dS)'  =  dx'dy'  +  dx'dz'  +  dy'dz'; 
dzV  ,    /dz 


'^«='i+iiij+fei  ^'""'^■ 


-=//(^+(iy+(i)')'-^-  ^-> 


INTEGRATION  AS  A  SUMMATION  OF  ELEMENTS.     237 

EXAMPLES. 
1.  Find  the  area  of  one  eighth  of  the  surface  of  the  sphere 

a;'  +  ^'  +  2"  =  a\ 

dz  X      dz  y 

Here  ^—  = ,     ^—  = . 

dx  z      dy  z 

1  -L  ^  ^  ^  =  1  4- ^  4_  ^  =  ^ 

^  dx-"^  dy""  ^  z'''^  z^       2*  * 

Substituting  in  (L),  we  have 

^  _    r  Padxdy  _      P  P        dxdy 


Va'  —  X-  —  y"" 

Integrating  first  with  reference  to  y  between  the  limits 
y  =  0  and  y  =  Va^  —  x',  we  get  the  differential  value  of  the 
element  B'C'KL;  and  then  integrating  with  reference  to  x 
between  the  limits  a;  =  0  and  a;  =  «,  we  get  the  sum  of  all  the 
elements  like  B'C'KL  between  these  limits,  which  sum  is  the 
area  required. 

Hence  S=a   I       /  f7a;rZ.y  ^nc^ 

Jq    Jo      Va"  -x"  —  y""         ^  ' 

2.  The  two  cylinders  x^  -\- z"^  =■  a^  and  x^  -\-  y"^  —  a^  intersect 
at  right  angles;  find  the  surface  of  the  one  intercepted  by  the 
other.  8«^ 


Here  z  =  ^0^  —  x^,  and  for  one  eighth  of  the  required  sur- 
face the  y-limits  are  0  and  l^a*  —  x^,  and  the  a;-limits  0  and  a. 
3.  A  sphere  whose  radius  is  a  is  cut  by  a  right  circular 

cylinder,  the  radius  of  whose  base  is  — ,  and  one  of  whose  edges 

passes  through  the  centre  of  the  sphere;  find  the  area  of  the 

surface  of  the  sphere  intercepted  by  the  cylinder.       2«^(7r  —  2). 

Take  x^  -\'  y"^  -\-  z"^  =  cC'  for  the  sphere,  and  x^  -{-y"^  ■=.  ax  ior 

the  cylinder,  then  z  =  Va^  —  y^  —  'x\  and  for  one  fourth  of  the 


238        DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 


required  surface  the  limits  of  y  and  x  are  0,  Vax  —  x",  and  0,  a, 
respectively. 

4.  In  the  preceding  example  find  the  surface  of  the  cylinder 
intercepted  by  the  sphere.     (See  Ex.  3,  Art.  233.)  4a^ 

5.  Find  the  area  of  the  portion  of  the  surface  of  the  sphere 
^^  4"  2/"  +  ^^  —  "^^y  lyiiig  within  the  paraboloid  y  =  mx^  +  ut^. 

In  a 


V 


mn 


237.  To  find  tlie  volume  of  a  solid  bounded  by  a  surface 
whose  equation  is  f{x,  y,  z)  =  0. 


Fig.  53. 


VOLUMES   OF   SOLIDS   DETERMINED  BY  TRIPLE  INTEGRATION. 

Let  V  =  the  indefinite  volume  expressed  by  the  product  of 
X,  y,  z\  then  v  =  xyz,  which  may  be  written 


V  =  I  I  j  dxdydz. 


which  becomes  definite  when  the  integrations  are  taken  between 
certain  limits,  and  we  will  now  give  the«  geometrical  interpreta- 
tion of  the  formula,  step  by  step. 


INTEGRATION  AS  A  SUMMATION  OF  ELEMENTS.     239 

Let  {x,  y,  z)  be  the  co-ordinates  of  tlie  point  P,  and  {x  -[■  dx, 
y  -\-  dy,  z  -\-  dz)  be  the  co-ordinates  of  the  point  Q;  then 

PQ  =■  dxdydz. 

(fl)  Eegarding  x  and  y  as  constant  and  integrating  between 
the  z-limits  0  and  id,  we  have 

'^\PQ'\  =  J^'  dxdydz  =  {id)dxdy  =  Ni. 

{b)  The  Tohime  of  bh  is  the  differential  value  of  the  element 
is'  with  respect  to  dy;  hence 

^""^  [is']  =  J^      {id)dx  dy  =  {afm)dx, 

which  is  the  volume  of  the  cylindrical  segment  afm-a'. 

(c)  The  volume  of  afm-a'  is  the  differential  value  of  the 
element  ar  with  respect  to  dx\  hence 

/■>0A 
{af'm)dx  —  volume  of  OBC-A. 

nOA     r*am,      nid 

Jo     Jo     Jo    ^^f^2/^^  =  volume  of  05(7-^.    .     .     (M) 

Cor.  I.  The  limits  of  y  and  z  are  found  thus:  id  is  the  posi- 
tive result  of  solving  the  equation /(a;,  y,  z)  =  0  for  z,  am  is  the 
positive  result  of  solving  f{x,  y,  0)  =  0  for  y,  and  OA  is  the 
positive  result  of  solving  f{x,  0,  0)  =  0  for  x. 

EXAMPLES. 
1.  Find  the  volume  of  one  eighth  of  the  ellipsoid 

a'  ^  b'  ^  c' 

I        x"^       ■?/'  \* 
The  limits  of  z  in  this  case  are  0  and  id=  c\\ ^  — 'jt]  5 


240  DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

the  limits  of  y  are  0  and  am  =  J(  1 ;;  I  ;  and  the  limits  of  x 

are  0  and  a.     Therefore  the  required  volume  is 

/  dx  dy  dz 

0      t/0 


6 

2.  Find  the  volume  of  the  solid  contained  between 
{a)  the  paraboloid  of  revolution,  x"  -\-  y""  =  az, 
{b)  the  cylinder,  x''  -{-  y''  =  2ax, 

(c)    and  the  plane,  z  =  0. 


dTta^ 


2    ■ 

X^  -\-  q/^ 

The   2;-limits  are  0  and    " — -^^,   the    ^/-limits  are  0   and 


V2ax  —  x",  and  the  a;-limits  are  0  and  2a,  for  one  half  of  the 
required  volume. 

3.  Eind  the  volume  cut  from  a  sphere  whose  radius  is  a  by  a 
right  circular  cylinder  whose  radius  is  5,  and  whose  axis  passes 
through  the  centre  of  the  sphere.  4;r  ,  3- 

^  -3-(a' -  (a^  -  5^r). 

4.  Find  the  entire  volume  bounded  by  the  surface  whose 

equation  is  x^  -\-  y^  -V  z^  =  «'.  ^^^ 

35    • 

5.  Find  the  volume  of  the  conoid  bounded  by  the  surface 
zV^  +  oj^y^  —  ^^^%  ^"d  the  planes  x  =  0  and  x  =  a.  ^nac^. 

APPLICATION  TO  MECHANICS. 
238.  Work  is  said  to  be  done  when  a  body  moves  through 
space  in  opposition  to  resistance.  A  horse  in  drawing  a  cart  or 
a  plough  does  a  certain  amount  of  work,  which  depends  on  the 
resistance  and  the  distance  traversed.  The  force  which  the 
horse  exerts,  and  the  distance  through  which  he  moves,  may  be 
regarded  as  the  two  elements  of  the  work  done.  If  r  lbs.  is  the 
constant  resistance  or  force,  and  x  feet  the  effective  distance 


INTEGBATION  A8  A  SUMMATION  OF  ELEMENTS.     241 


through  which  the  body  moves,  rx  units  of  work  will  be  done. 
By  effective  distance  is  meant  the  distance  measured  in  the 
same  direction  as  that  in  which  the  force  is  acting.  Thus,  when 
the  resistance  is  constant,  the  amount  of  work  may  be  represented 
by  the  area  of  a  rectangle  whose  base  is  the  distance  {x)  and 
whose  altitude  is  the  resistance  (r). 

If  the  resistance  or  force  is  a  variable  dependent  on  the  dis- 
tance X,  it  may  be  represented  \)jf'{x),  in  which  case  the  amount 
of  work  may  be  found  by  taking  the  sum  of  its  elements,  thus: 
In  Fig.  43,  if  the  force  f'{x)  (=  BP)  act  through  the  small 
effective  distance  h  (=  BC),  the  work  done  will  be  in  excess  of 
f'{x)h  (=  BCDP)  only  by  the  acceleration  of  work  {PDF') 
during  that  interval.  Hqucq,  f ' {x)dx  is  the  differential  value  of 
the  xt}\  element  of  work.  Therefore  the  quantity  of  work 
between  the  limits  x  =^  x^  and  x  =  x^  is,  viz. : 

^l\E^'\or^^l2^^y\x)h-]=£y'{x)dx,      .     (N) 

in  which  the  effective  distance  is  x^  —  x^. 

Cor.  I.  Effective  distance,  resistance,  and  work,  and  effective 
distance,  force,  and  energy,  bear  the  same  relation 
to  one  another  as  the  abscissa,  ordinate,  and  area  of 
a  plane  curve  referred  to  rectangular  co-ordinates, 
respectively. 

Example.  Let  it  be  required  to  compute  the 
quantity  of  work  necessary  to  comj)ress  the  spiral 
spring  of  the  common  spring-balance  to  any  given 
degree,  say  from  AB  to  DB.'^ 

Let  the  resistance  (=:/'(.-c))  vary  directly  as 
the  degree  of  compression,  and  denote  the  distance 
AD'  by  x;  then  will 


f'{x)  =  7nx, 


Fig.  54. 


where   m   is   the  resistance   of   the  spring  when  the  balance 


Bartlett's  Analytical  Meclianics,  page  39. 


242         DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

is  compressed  through  the  distance  unity.    Substituting  in  (N), 
making  x^  =  0  and  x^  =  AD,  we  have 


the  work  =   }     mx  dx  = 

^0 


T+c 


imx„''. 


CoE.  I.  If  m  =  10  pounds  and  x^  =  3  ft.,  then  will 

the  work  =  45  units  of  work; 

that  is,  the  quantity  of  work  will  be  equal  to  that  required  to 
raise  45  pounds  through  a  vertical  height  of  one  foot. 

CENTRE  OF  GRAVITY. 

239.  The  bodies  here  considered -are  supposed  to  be  of  uni- 
form density;  that  is,  equal  quantities  of  a  body  have  equal 
weights. 

The  Centre  of  Gravity  of  a  body  is  a  point  so  situated  that 
the  force  of  gravity  produces  no  tendency  in  the  body  to  rotate 
about  any  axis  passing  through  this  point. 

The  Moment  of  any  element  or  particle  of  a  body  with 
reference  to  any  horizontal  axis  is  the  product  of  the  magnitude 
or  weight  of  the  element  by  the  horizontal  distance  of  its  centre 
of  gravity  from  the  axis^  and  measures  the  tendency  of  the 
element,  under  the  influence  of  gravity,  to  produce  rotation 
about  the  axis.  The  moment  of  the  body  itself  is  the  sum  of 
the  moments  of  its  elements.  If  the  axis  of  reference  passes 
through  the  centre  of  gravity  of  the  body,  the  moment  of  the 
body  must  be  zero,  otherwise  the  moments  of  the  elements  would 
not  neutralize  one  another,  and  the  body  would  rotate. 

240.  To  find  the  centre  of  gravity  of  a  plane  area. 

In  Fig.  43,  suppose  the  plane  curve  placed  in  a  horizontal 
position,  and  let  A  =  the  area  of  EGQF,  x  =  OB,  y  =  BP, 
x^  —  OE,  x^=^  00.  Also  let  (x' ,  y')  be  the  centre  of  gravity 
of  A,  and  [x  -\-  a,  y  -\-  ^)  be  the  centre  of  gravity  of  the  rect- 
angle BCDP  {=  yh).  Evidently  the  limit  of  a,  as  BO  or  h 
approaches  0,  is  0. 


INTEQBATION  AS  A  SUMMATION  OF  ELEMENTS.     243 

The  moment  of  BCDP  with  respect  to  an  axis  passing 
through  {x',  y')  and  parallel  to  the  axis  of  y  is  (.r  +  «  —  x')yl/, 
which  is  the  measure  of  the  tendency  of  this  rectangle  (BCDP) 
to  produce  rotation  about  the  given  axis,  and  therefore  the 
tendency  of  all  the  similar  rectangles  to  produce   rotation   is 

2  ^[x -^  a  —  x'^yJt .      The   smaller   the    rectangles  the  nearer 

their  sum  comes  to  the  whole  area  of  the  curve,  and  therefore 
the  tendency  of  A  to  rotate  about  the  given  axis  is 


limit 
h  or  a 


.   r^^'  \x  +  «-  —  x')yh  =   /     (x  —  x')ydx; 
=  0    x,  «/a-, 


but  as  the  axis  of  reference  passes  through  the  centre  of  gravity 
of  A,  this  must  equal  zero. 

/     {x  —  x')ydx  =    /     xydx  —  x'  I  ydx  —  0. 

/•j-2  rx^  pxi 

Whence    x'  =    I     xydx  -~    /     ydx  =    I     xydx  -r-  A.     (P) 

X\  X  J  X\ 

In  like  manner  we  find 

y'  =  f^^'fdx  ^A (Q) 

2i4:\.  To  find  the  centre  of  gravity  of  a  plane  curve- 
In  Fig.  44,  suppose  the  plane  curve  PQ  {=  s)  placed  in  a 
horizontal    position;     let    x  =  OB,    y  =  BP,   x^  =  OG,   and 
h  =  BC;  also  let  [x',  y')  be  the  centre  of  gravity  of  s,  and 
{x  -\-  a,  y  -\-  ^)  the  centre  of  gravity  of  the  tangent 


'i='^'^^(m- 


\dx , 
The  limit  of  a,  as  h  approaches  0,  is  evidently  0. 


244         DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

The  moment  of  Pt  with  respect  to  an  axis  passing  through 
{x' ,  y')  and  parallel  to  the  axis  of  y  is  {x-\-a—x')A/\   i    (^ ]  Ji 

and  the  sum  of  the  moments  of  all  the  tangents  like  Pt  is 

X  id  i/^\  i 

^  \x  -{-  a  —  x'][l-\-  -f^]  h.    The  limit  of  this  sum,  as  h  and  a 

approach  0,  is  equal  to    /     (a;  —  x')ds,  which  is  the  moment  of 
s  with  respect  to  the  given  axis,  since  s  =  ,  _^  ^    [^^]- 

/     (x  —  x')ds  =   I     xds  —  x'  /  ds  =  0. 

f*Xi 

Whence  x'  =^    I      xds  -r-  s (E) 

tJx 

In  like  manner  we  find 

y  =         yds--  s (S) 

242.  Let  the  student  prove  in  a  similar  manner  that  the 
formula  for  finding  the  centre  of  gravity  of  a  solid  of  revolution 
whose  axis  is  the  axis  of  x  is 


x'  :=  TT  /    ^xy'^dx  -^  the  volume; 
or  x'  =  I  ^'^xy'^dx  -^   /    y'^dx. 

d  Xx  dxi 


y     .    .    .     (T) 


Note. — As  the  centre  of  gravity  must  evidently  be  on  the 
axis  of  revolution,  the  formula  given  above  entirely  determines 
it.     The  same  is  true  of  the  following  formula. 

Prove  that  the  formula  for  finding  the  centre  of  gravity  of 
any  surface  of  revolution  whose  axis  is  the  axis  of  x  is 

x'  =         \-i/ds  -^   /    ^y(Ts (U) 


INTEGRATION  AS  A  SUMMATION  OF  ELEMENTS.    246 
EXAMPLES. 

1.  Determiue   the    centre    of    gravity    [G)   of  an  isosceles 

triangle. 

Let  OD,  the  altitude,  =  a,  DC,  half  of  the  base, 

Ix 
=  h,  OB  —  X,  BP  =  y,  then  y  =  —,  and,  by   for- 


mula (P), 


/    hx'' 


area  ADC  iab 


OG  = TT^  =  ^^^^^ =  f«j 


that  is,  the  distance  of  the  centre  of  gravity  from  the  vertex 
of  the  triangle  is  equal  to  two  thirds  of  the  altitude  of  the 
triangle. 

3.  Determine  the  centre  of  gravity  of  the  area  bounded  by 
the  parabola  if  =  iax  and  the  double  ordinate  (2y)  perpendicular 
to  the  axis  of  x.  x'  =  ^x,  y'  —  0. 

3.  Find  the  centre  of  gravity  of  the  area  of  the  curve 
xy""  =  ¥{a  —  x).  x'  =  ia,  y'  =  0. 

4.  Find  the  centre  of  gravity  of  the  area  of  the  cycloid. 

x'  =  7tr,  y'  =  |r. 

5.  Find  the  centre  of  gravity  of  the  area  of  x^  -\-  y^  =  a^ 
lying  in  the  first  quadrant.  ,  _    ^  _  256  a 

""  -y  -  315  ^• 

6.  Find  the  centre  of  gravity  of  the  area  of  a'^y''  -f-  6 V  =  a'V, 
lying  in  the  first  quadrant.  ,  _  4a      ,  _  46 

''^  ~  3^'  ^   =  3^- 

7.  Find  the  centre  of  gravity  of  the  arc  of  the  circle  x"^  -f  y"^ 
—  r"  lying  in  the  first  quadrant.     (See  formulas  (R)  and  (S).) 

2r 

X   z=  y'  =z  —  . 

8.  Find  the  centre  of  gravity  of  the  arc  of  the  curve  x'-{-y^ 
=  a^  lying  in  the  first  quadrant.  x'  =  y'  —  |a. 


246         DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 
9.  Find  the  centre  of  gravity  of  the  arc  of  the  cycloid 

y 


X  =■  r  vers~^  -  —  V^ry  —  y'^, 

lying  between  (0,  0)  and  {nr,  2r).  x'  =  |r,  y'  =i  -\-  ^r. 

10.  Find  the  centre  of  gravity  of  the  paraboloid  generated 
by  revolving  y^  =  4mx  about  the  axis  of  x.     (See  formula  (T).) 

11.  Find  the  centre  of  gravity  of  the  segment  of  a  sphere 
generated  by  revolving  t/^  =  2rx  —  x'^  about  the  axis  of  x. 

,  _  x{Sr  —  3x) 
~  4:{3r  —  x)  ' 
When  X  =  r,  x'  =  |r. 

12.  A  semi-ellipsoid  is  formed  by  the  revolution  of  a  semi- 
ellipse  about  its  major  axis;  find  the  distance  of  the  centre  of 
gravity  of  the  solid  from  the  centre  of  the  ellipse.  x'  =  fa. 

13.  Find  the  centre  of  gravity  of  the  convex  surface  of  the 
cone  generated  by  revolving  the  line  y  =  mx  about  the  axis  of  x. 
(See  formula  (U).)  a^'  =  fa;. 

14.  Find  the  centre  of  gi'avity  of  the  surface  of  a  spherical 
segment  v^^hose  altitude  is  x.  x'  =  ^x. 

15.  Find  the  centre  of  gravity  of  the  surface  of  the  parabo- 
loid generated  by  revolving  y'^  =  Amx  about  the  axis  of  x. 

,  _  1  (3a;  —  2m){x  -f  mf  +  2w^  ^ 

'^  —  K  'i  \^        ^  * 

•^  {x  -\-  7nr  —  irf 


APPENDIX. 


Aj.  Differentiable  Functions.  A  function,  y  =  f{x),  is 
said  to  be  differentiable  when  -~  approaches  a  definite  limit  as 

Jx   approaches   zero.      Thus,  y  =  Vx  is   differentiable,  since 
(Art.  10,  ex.  5) 

^x      \/x'  +  A  +  Vx' 

approaches  the  definite  limit,  — — ,  as  h  approaches  zero,  x' 

being  any  particular  or  definite  value  of  x  from  which  h  is 
estimated. 

All  ordinary  continuous  functions  are  differentiable,  but  this 
does  not  follow  from  the  mere  fact  that  the  functions  are  con- 
tinuous, for  there  are  functions  which  are  continuous  and  yet 
have  no  differential  coefficients.*  Functions  of  this  kind,  how- 
ever, are  of  such  rare  occurrence  that  the  distinction  between 
continuity  and  differentiability  is  seldom  made  in  works  on  the 
Differential  Calculus.  That  is,  every  function  is  regarded  as 
continuous  and  differentiable  between  certain  limits. 

/ill 
The  limit  {ni^  of  -p  in  any  particular  case  can  often  be 

conveniently  determined  by  assuming  that  Ay  =  mji  +  mji^, 

*  See  Harkness  and  Morley's  Theory  of  Functions. 

247 


248  APPENDIX. 

which  is  true  of  all  differentiable  functions  of  a  single  variable, 
and  then  finding  the  value  of  m^ ,  as  in  A^.  The  general 
values  of  m^  and  m^,  and  the  exact  conditions  under  which 
/}y  z=  mji  +  mji'  holds,  are  given  in  A,. 

Ag.  Another  Illustration  of  the  Formula  Ay  =  mji  +  mji\ 
Suppose  that  a  moving  body  has  traversed  a  distance  (s)  in  the 
time  t,  and  that  the  value  of  s  in  terms  of  t  is 

«=/(0 (1) 

Suppose  we  wish  to  find  the  actual  velocity  (vj  at  the  end  of 
the  time  t^.  Let  /It,  an  increment  of  f  estimated  from  t^,  be 
any  arbitrary  period  immediately  succeeding  the  end  of  the  time 
?^j,  then  the  distance  traversed  by  the  body  in  that  period  Avill 
be  the  corresponding  increment  of  s,  viz., 

/Is  =f{t  +  At)  -  f{t)  =  m^At  +  m^{Aty.    .     .     (2) 

The  mean  velocity  [v)  of  a  moving  body,  during  any  period 
of  time,  is  the  quotient  obtained  by  dividing  the  distance 
traversed  by  the  body  by  the  length  of  the  period.  Therefore 
the  mean  velocity  during  the  period  At  is 

As 

—  =  m^  +  m^At  =  v (3) 

Now  this  mean  velocity  evidently  approaches  the  actual 
velocity  v^  as  At  aj)proaches  0,  indefinitely.  Hence  taking  the 
limit  of  (3),  we  have 

limit  of  —r,  =  m,  =  v.. 
At         '        ' 

A,<i 
That  is,  the  limit  of  -r  ,  or  m, ,  is  the  actual  velocity  of  the 

body  at  the  end  of  the  time  t^ ,  and  hence  m^At  is  what  As  would 
have  been  had  it  varied  as  At  or  had  the  actual  velocity  v^  re- 
mained constant,  and  mji^  is  the  acceleration  of  s  during  tlie 
period  At. 


APPENDIX.  249 

Ag,  The  Differentials  of  Independent  Variables  are,  in 
general,  Variables.  In  differentiating  y  =.  f[x)  successively 
dx  is  usually  regarded  as  a  constant;  that  is,  as  having  the  same 
value  for  all  values  of  x.  "  This  hypothesis/^  say  Eice  and  John- 
son,* "greatly  simplifies  the  expressions  for  the  second  and 
higher  differentials  of  functions  of  x,  inasmuch  as  it  is  evidently 
equivalent  to  making  all  differentials  of  x  higher  than  the  first 
vanish."  Again,  "  A  differential  of  the  second  order  or  of 
a  higher  order,"  says  Byerly,f  "  has  been  defined  by  the  aid  of  a 
derivative,  which  always  implies  the  distinction  between  func- 
tion and  variable,  and  on  the  hypothesis  of  an  important  dif- 
ference in  the  natures  of  the  increments  of  function  and  vari- 
able; namely,  that  the  increment  of  the  independent  variable  is 
a  constant  magnitude,  and  that,  consequently,  its  derivative  and 
differential  are  zero." 

The  impressions  which  these  and  similar  statements  in  other 
excellent  works  are  likely  to  make  on  the  mind  of  the  student 
are  {a)  that  all  independent  variables  vary  uniformly,  and  {h) 
that  they  must  vary  in  this  manner  in  order  that  the  differen- 
tials of  their  differentials  shall  be  zero. 

That  an  independent  variable  may  vary  uniformly,  as  in  Eate 
of  Change,  is  granted;  but  differentials  in  general  are  variables 
whose  limits  are  zero.  Indeed  one  of  the  most  important  and 
essential  properties  of  the  differential  of  an  independent  vari- 
able is  its  independent  variability.  The  imposition  of  any  con- 
dition on  a  group  of  variables  by  which  they  may  be  expressed  in 
terms  of  one  another  at  once  destroys  the  independence  of  the 
variables,  and  this  is  the  case  of  variables  under  the  hypothesis 
of  uniform  variation,  or  rate  of  change. 

Thus,  let  u  =  f{x,y),  and  let  us  suppose  x  and  y  to  vary 
simultaneously  and  uniformly;  then  dx  =  mdt  and  dy  =  m'dt; 
whence  x  =  mt  -\-  O  and  y  —  m't  -\-  C.  Eliminating  t  and 
solving  for  y,  we  have  y  =  (p{x).  Hence  the  supposition  of 
uniform  change  renders  the  hypothesis  of  more  than  one  in- 

*  Dif .  Calculus,  Art.  79.  f  Dif .  Calculus,  Art.  204. 


250  APPENDIX. 

dependent  variable  impossible.  Therefore,  if  independent 
variables  (which  are  supposed  to  vary  simultaneously)  must  vary 
uniformly  in  order  that  their  higher  differentials  shall  be  zero, 
the  successive  differentials  of  u  =  f{y,  x,  z)  can  be  obtained 
only  by  destroying  the  independence  of  all  the  variables  x,  y,  z 
except  one. 

Hence,  in  general,  if  the  differential  of  dx,  with  respect  to 
X,  is  zero,  it  is  due  to  the  fact  that  dx  is  a  variable  which  is 
independent  of  x. 

A^.  To  differentiate  a^'  and  log„  v  independently  of  Art. 
78. 

Let  y  =  a^,  where  v  is  a  function  of  x. 

Increasing  x  by  li,  etc.,  and  assuming  that  cf  is  differentiable 
{A^,  we  have 

Ay  la^-"  —  1\  ,         . 

Av  \    Av    I  1  1      'i 

Taking  the  limits,  remembering  that  m^Av  vanishes  with  lu 
and  that  a"  is  constant  with  respect  to  h,  we  have 


dv  \Av^O 


11 


Av     J 


=  w^, 


where  m^  and  a"  are  definite  quantities.     Therefore  the  limit  of 

— -j must  be  a  definite  quantity  {m'  say),  not  zero,  and  de- 

Av 

pendent  solely  on  the  base  a,  since  it  is  evidently  independent 

of  X  and  h. 

dv 

~r^  =  oym' ,     or     dy  =  a^'m'dv (1) 

dv  ^  ^  ' 

Cor.  I.  Since  m'  depends  on  the  base  and  the  base  is  arbi- 
trary, Ave  may  suppose  the  base  to  have  s,uch  a  value  (say  e)  that 
vi'  —  1.     Then  de"  =  e'-'dv. 


APPENDIX.  251 

CoE.  II.  To  find  the  value  of  m'  in  (1),  let 

a-"  =  e"" (2) 

Differentiating  (2),       a^'m'dv  =  e^du (3) 

fill 
From  (2)  and  (3),  ^'  =  7~ (4) 

Taking  loge  of  (2),       v  log^  a  ^  u (5) 

Differentiating  (5),    dv  log^  a  =  du (6) 

Whence  loge  ^  =  ~r~ ^^^ 

Equating  (4)  and  (7),  m'  —  loge  (f (8) 

d{a'')  =  a"  loge  adv (9) 

Cor.  III.  To  differentiate  i/  =  log^  v,  we  write  it  under  the 
form  of 

a^  =  v (10) 

Differentiating  (10),     a^  loge  <^<'(^y  =  ^^^5 

whence  dy  =  , or    log„  s — (11) 

•^       loge  a  V  "="     V  ^     ^ 

A^.  A  Rigorous  Proof  of  Taylor's  Formula.  In  what  fol- 
lows, the  function  f{i/)  and  its  n  successive  derivatives  are  sup- 
posed to  be  differentiable,  and  finite  and  continuous  between 
the  limits  y  and  y  -{-  x. 

Lemma.  If  F{z)  is  continuous  betwen  «  =  a  and  z  =^  y,  and 
if  F{a)  —  F{y)  =  0,  then  F'{z),  if  continuous,  must  equal  zero 
for  some  value  of  z  between  a  and  y. 

For,  as  z  changes  from  a  to  y,  F{z)  passes  from  0  to  0,  that 
is,  F{z)  increases  and  then  decreases,  or  vice  versa;  hence  F'{z) 
must  change  from  +  to  —  or  from  —  to  -\-,  and  therefore,  since 
it  is  continuous,  pass  through  0. 


252  APPENDIX. 

lu  what  follows  6  v/ill  represent  a  positive  proper  fraction; 
that  is,  0  <  /^  <  1.  Hence,  0  <  dx  <  x,  and,  by  the  lemma, 
F'[y  +  B{a  -  y)-]  =  0. 

Under  the  given  hypotheses,  we  have  (Aj) 

r~\y  +  ^)  =f''~\y)  +  m^x  +  m^x\ 

or  f-^{y  +  x)=r-\y)  +  xs, (1) 

where  s  (=  m^  -\-  m^x)  is  continuous  between  y  and  y  -\-  x. 

Multiply  (1)  by  dx,  regard  y  constant,  and  integrate,  and  we 
have 

r-\y  +  ^)=  r-\y)  +  ^r'\y)  +  f^s  dx,  .    .    (2) 

the  constant  C  being  f"~^{y),  since /"^'^{y  -\-x)  =.  f-'^[y)  when 
a;  =  0. 

Multiply  (2)  by  dx,  and  integrate,  denoting  /  /  by  /  , 
and  we  have 

.r-\y  4-  X)  =r-\y)  +  :^r-\y)  +  jr-\y)  +  f'xs  dx\   (3) 

C  being /^-«(2/). 

Continuing  thus  to  w  —  1  integrations,  we  have 

Ay  +  ^)  =  M  +  «/'(.^)  +  -^/"(y) 

-V...y^r-\y)-^fxsdx--\    (4) 

/n— 1 

which  is  the  remainder  after  n  terms. 

In  (4)  put  a  —  y  for  x,  and  s'  for  the  corresponding  value  of 
s,  transpose,  and  we  have 

/(«)-/W-^/'W-^/"W 

- . . .  ^^-^r-\y)  -  f^\-y)s  {-dyY-^=o.    (5) 


APPENDIX.  353 

Let  F{z)  be  a  function  such  that 

F{z)  =  f{a)  -m  -  ^/'(^)  -  ^^^r'^ 

_  .  .  .  (filli^y-i(^)  _  J^'^  \a-z)s'{-dzY-' ...     (6) 

Since  *"',  having  the  same  Talue  as  in  (5),  is  independent  of 

(fi 2;)"' 

z,  the  last  term  of  (6)  is  equal  to j s\ 

Evidently  F{z)  =  0,  first  when  z  =  2/  hj  (5),  second  when 
z  =  a;  and  since  f{y),f'{y),f"{y),  etc.,  are  all  continuous  from 
y  to  y  -{-  X  {=  a),f{z),f'{z),f"{z),  etc.  (and  therefore  F{z)  and 
F'{z) ),  are  continuous  between  the  same  limits.  Therefore,  by 
the  lemma,  F'{y  +  0{a  -  y))  =  0. 

Differentiating  (6)  to  obtain  F'{z),  we  have 

F\z)  =  -f\z)  +r{z)  -  ^^^/-(.)  +  ^/"(^) 

That  is,  i^'(^)  =  -  ^-^^^W)  -  ^']-    •••(') 


Now  substituting  y  +  6{a  —  y)  for  z,  observing  that  for  this 

(a  —  zY-^ 
value  of  z,  F'{z)  =  0,  and  dividing  by     ,  ^  _  -^^    ,  we  have 

5'=/%  +  %-^)] (8) 

In    (6)    substitute  this  value  of  s',  and  then  put  x  for  2; 


254  APPENDIX. 

(whence  F{z)  =  0),  y  -{-  x  for  a  (wlience  a  —  y  =  x),  transpose, 
and  we  have 

f\y  +  ^)  =f{y)  +  ^f{y)  +  wJ"iy) 


+  •  •  •  u^rir^f'-'iy)  +  7^/"(?/  +  ^^)-   (9) 


The  last  term 


-urfiy  +  ^^)  =  Rn,  say. 


is  called  the  remainder  in  Taylor's  formula.  It  is  obtained  by 
differentiating  f[y)  n  times  and  substituting  ?/  +  6x  for  y  in  the 
final  or  nt\\  derivative. 

When  the  function /(;?/  +  ^')  is  such  that  this  remainder  ap- 
proaches 0  as  n  approaches  co ,  the  series  will  be  convergent, 
otherwise  it  will  be  divergent.  Hence  the  remainder  enables  us 
to  ascertain  the  conditions  under  which  any  given  function  of 
the  sum  of  two  variables  is  developable  by  Taylor's  formula, 
and  also  to  find  the  limits  of  the  error  we  make  in  stopping  at 
any  term  of  the  series. 

EXAMPLE. 

1.  f{y  +  x)=  log  {y  +  x).     See  Art.  125. 

\n-l 


Since        f{y)  =  log  y,    f"{ij)  =  -  {- 1)\ 


'   n  \ 


n  \y  +  Ox 


Now,  since  0  <  6'  <  1,  —  is  finite  and  a  proper  fraction 

y  -\-  ux 

if  X  =  or  <  y.     Therefore  i?„  approaclxes  0  as  n  approaches  oo , 
and  log  {y  -\-  x)  is  developable  if  a;  =  or  '<  y. 


APPENDIX.  255 

Agaii],  since  0  <  6^  <  1,  the  true  numerical  value  of  i?„  lies 

between—  -     and  -  — '- — ).     Hence   if  —  (— 1)"— ( -)     be 
n  \yi  ■)i  \y  -\-  x/  '  n  \y ) 

substituted  for  R^  in  (1),  Art.  125,  the  series  will  be  the  value 

1  /      X 
of  log  {y  +  x)  to  within  less  than  — 


n\y  +  XI 

A^.  A  Similar  Completion  of  Maclaurin's  Formula  may  be 
obtained  by  making  ?/  =  0  in  (9),  A^.     Thus, 

AA  =/(0)  +  i/'(0)  +  j^/-"(o) 

+  ---^/-(0)+|/-(&)-   (1) 

EXAMPLE. 

/(:r)  =  e. 

Since  Mx)  =  e,  i?„  =  -|^e^^. 

For  any  finite  value  of  x,  (1)  the  fraction  —  evidently  ap- 
proaches 0  as  n  approaches  co ,  and  (2),  since  0  <  ^  <  1,  e^-^  is 
finite.  Hence  the  limit  of  i?„ ,  as  n  approaches  oo ,  is  0,  and  (f 
is  developable,  for  all  finite  values  of  x. 

Again,  as  0  <  6^  <  1,  the  true  value  of  7?,.  lies  between  - — 

\n 

x^' 
and  -; — <3^.     Therefore  the  sum  of  all  the  terms  after  the  nth.  in 

In 

a;" 
series  (N),  Art.  127,  is  less  than  - — e"". 

\n 

A,.  To  determine  the  values  of  m^  and  m^  in  the  formula 

Jy  =  mji  +  mji'  (Art.  24). 

Since  y  =  f{x),  Ay  =  fix  +  It)  -  f{x). 


256  APPENDIX, 

By  Taylor's  formula, 

f{:x  +  h)  =/(^)  +  ¥'(^)  +  '^rix  +  eii). 

Ay  =.  f'{:x)h  +  ^f'\x  +  e]i)li\ 

Comparing  this  with  the  given  formula,  we  have  w?j  —f'{x) 
and  ?;?,  =  |-/"Ci-  +  6^/0,  where  0  <  ^  <  1. 

Therefore  the  formula  Ay  =  mji  -\-  m.Ji'  is  true  in  reference 
to  y  =  f{x),  (1)  if  /(?;)  and  f'{x)  are  differentiable,  and  '(2)  if 
f{x),  f'{x),  and  f"{x)  are  finite  and  continuous  between  the 
limits  X  and  x  -j-  h. 


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